= 72 \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Expand the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 6x + 12 = 72 \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Isolate the term with \\( x \\) by subtracting 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 6x = 60 \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Solve for \\( x \\) by dividing both sides by 6: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 10 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe answer is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 10 \u003C/math-field>\u003C/math-field>",380,76,"write-and-solve-an-equation-find-a-number-such-that-tripling-the-sum-of-twice-the-number-and-4-is-72",{"id":36,"category":28,"text_question":37,"photo_question":30,"text_answer":38,"step_text_answer":8,"step_photo_answer":8,"views":39,"likes":40,"slug":41},537924,"Write and solve an equation:\nFind a number such that 5 less than the opposite of the number is 3 times the sum of the number and -7","1. Write the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-x - 5 = 3(x - 7)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Expand the right side: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-x - 5 = 3x - 21\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> to both sides: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-5 = 4x - 21\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Add 21 to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>16 = 4x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Divide by 4:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 4\u003C/math-field>\u003C/math-field>",1033,207,"write-and-solve-an-equation-find-a-number-such-that-5-less-than-the-opposite-of-the-number-is-3-times-the-sum-of-the-number-and-7",{"id":43,"category":28,"text_question":44,"photo_question":30,"text_answer":45,"step_text_answer":8,"step_photo_answer":8,"views":46,"likes":47,"slug":48},537923,"Find two consecutive odd integers whose sum is 116","**\u003Cbr />\n\u003Cbr />\n1. Set up the equation based on the problem statement:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + (x + 2) = 116 \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Simplify by combining like terms:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x + 2 = 116 \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Subtract 2 from both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x = 114 \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Divide both sides by 2 to solve for \\( x \\):\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 57 \u003C/math-field>\u003C/math-field>\u003Cbr />\n5. The first odd integer is 57, the next consecutive odd integer is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + 2 = 59 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the two consecutive odd integers are \\( x = 57 \\) and \\( x + 2 = 59 \\).",1099,220,"find-two-consecutive-odd-integers-whose-sum-is-116",{"id":50,"category":28,"text_question":51,"photo_question":30,"text_answer":52,"step_text_answer":8,"step_photo_answer":8,"views":53,"likes":54,"slug":55},537922,"Find three consecutive integers whose sum is 84","1. Represent the consecutive integers as $x$, $x+1$, and $x+2$.\u003Cbr />\n\u003Cbr />\n2. Write the equation for their sum: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + (x+1) + (x+2) = 84\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify and solve for $x$:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 3 = 84\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x = 81\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 27\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The integers are $x = 27$, $x + 1 = 28$, $x + 2 = 29$.\u003Cbr />\n\u003Cbr />\n5. Therefore, the three consecutive integers are 27, 28, and 29.",1301,260,"find-three-consecutive-integers-whose-sum-is-84",{"id":57,"category":28,"text_question":58,"photo_question":30,"text_answer":59,"step_text_answer":8,"step_photo_answer":8,"views":60,"likes":40,"slug":61},537921,"A pilot flew 390 nautical miles in 45 minutes. Find his average speed in knots (nautical miles per hour)","1. Convert time from minutes to hours: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\text{{Time in hours}} = \\frac{45}{60} = 0.75 \u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. Calculate average speed: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\text{{Average speed}} = \\frac{390}{0.75} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n3. The average speed of the pilot is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 520 \u003C/math-field>\u003C/math-field> knots.",1037,"a-pilot-flew-390-nautical-miles-in-45-minutes-find-his-average-speed-in-knots-nautical-miles-per-hour",{"id":63,"category":28,"text_question":64,"photo_question":30,"text_answer":65,"step_text_answer":8,"step_photo_answer":8,"views":66,"likes":67,"slug":68},537920,"Write and solve an equation:\nHow long does it take Kathryn to get to her grandmother’s house that is 180 mi away if she averages 45mi/hr on the trip","1. Use the formula for time: \\( t = \\frac{d}{s} \\).\u003Cbr />\n2. Plug in the distance: \\( d = 180 \\, \\text{miles} \\).\u003Cbr />\n3. Plug in the speed: \\( s = 45 \\, \\text{miles per hour} \\).\u003Cbr />\n4. Solve for time: \\( t = \\frac{180}{45} = 4 \\).\u003Cbr />\n5. Answer: \\( 4 \\, \\text{hours} \\).",668,134,"write-and-solve-an-equation-how-long-does-it-take-kathryn-to-get-to-her-grandmother-s-house-that-is-180-mi-away-if-she-averages-45mi-hr-on-the-trip",{"id":70,"category":28,"text_question":71,"photo_question":30,"text_answer":72,"step_text_answer":8,"step_photo_answer":8,"views":73,"likes":74,"slug":75},537919,"Write and solve an equation:\nFind a number such that the difference of 97 and eighteen times the number is five times one more than the number","1. Let the number be \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. According to the problem, the difference of 97 and eighteen times the number (\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18x \u003C/math-field>\u003C/math-field>) is equal to five times one more than the number (\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 5(x+1) \u003C/math-field>\u003C/math-field>).\u003Cbr />\n\u003Cbr />\n The equation can be written as: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 97 - 18x = 5(x + 1) \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Expand the term on the right-hand side: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 5(x + 1) = 5x + 5 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Substitute back into the equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 97 - 18x = 5x + 5 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Bring all terms involving \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \u003C/math-field>\u003C/math-field> to one side and constant terms to the other:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 97 - 5 = 18x + 5x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Simplify both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 92 = 23x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n7. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{92}{23} = 4 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the number is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 4 \u003C/math-field>\u003C/math-field>.",1373,275,"write-and-solve-an-equation-find-a-number-such-that-the-difference-of-97-and-eighteen-times-the-number-is-five-times-one-more-than-the-number",{"id":77,"category":28,"text_question":78,"photo_question":30,"text_answer":79,"step_text_answer":8,"step_photo_answer":8,"views":80,"likes":81,"slug":82},537918,"Find two consecutive integers such that four times the larger is 94 more than three times the smaller","1. Let the two consecutive integers be \\( x \\) and \\( x + 1 \\).\u003Cbr />\n2. According to the problem, four times the larger integer is 94 more than three times the smaller integer. Therefore, the expression becomes: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4(x + 1) = 3x + 94 \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Distribute and simplify: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4x + 4 = 3x + 94 \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Subtract \\( 3x \\) from both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + 4 = 94 \u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Subtract 4 from both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 90 \u003C/math-field>\u003C/math-field>\u003Cbr />\n6. The larger integer is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + 1 = 90 + 1 = 91 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nAnswer: The larger integer is 91, so the consecutive integers are 90 and 91.",1053,211,"find-two-consecutive-integers-such-that-four-times-the-larger-is-94-more-than-three-times-the-smaller",{"id":84,"category":28,"text_question":85,"photo_question":30,"text_answer":86,"step_text_answer":8,"step_photo_answer":8,"views":87,"likes":88,"slug":89},537917,"If 30% of the students in a class made a B and there were 12 Bs how many students were in the class?","1. Let the total number of students in the class be denoted by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. According to the problem, 30% of the students made a B, and this equals 12 students.\u003Cbr />\n3. Set up the equation:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.30 \\times x = 12\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. To find \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>, divide both sides of the equation by 0.30:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{12}{0.30}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify the right side:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{12}{0.30} = 40\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Therefore, the total number of students in the class is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\boxed{40} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nAnswer: 40 students.",962,192,"if-30-of-the-students-in-a-class-made-a-b-and-there-were-12-bs-how-many-students-were-in-the-class",{"id":91,"category":28,"text_question":92,"photo_question":30,"text_answer":93,"step_text_answer":8,"step_photo_answer":8,"views":94,"likes":95,"slug":96},537916,"Evaluate \\int_1^4 (5 - 2x + 3x^2) dx :","Solution:\u003Cbr />\n1. Identify the function to integrate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f(x) = 5 - 2x + 3x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Integrate each term:\u003Cbr />\n- The integral of a constant \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> with respect to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5x\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- The integral of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2x\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-\\frac{2}{2}x^2 = -x^2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- The integral of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x^2\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{3}{3}x^3 = x^3\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Combine the integrals:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int (5 - 2x + 3x^2) \\, dx = 5x - x^2 + x^3 + C\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Evaluate the definite integral from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>:\u003Cbr />\n- Compute the antiderivative at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F(4) = 5(4) - (4)^2 + (4)^3 = 20 - 16 + 64 = 68\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Compute the antiderivative at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F(1) = 5(1) - (1)^2 + (1)^3 = 5 - 1 + 1 = 5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find the difference \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F(4) - F(1)\u003C/math-field>\u003C/math-field>: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>68 - 5 = 63\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. The value of the integral is:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int_1^4 (5 - 2x + 3x^2) \\, dx = 63\u003C/math-field>\u003C/math-field>",1389,278,"evaluate-int-1-4-5-2x-3x-2-dx",{"id":98,"category":28,"text_question":99,"photo_question":30,"text_answer":100,"step_text_answer":8,"step_photo_answer":8,"views":101,"likes":102,"slug":103},537915,"Find the area between y = 9 - x^2 and y = x + 1 from x = -1 to x = 2","Solution:\u003Cbr />\n\u003Cbr />\n1. Given:\u003Cbr />\n- Curve 1: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 9 - x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Curve 2: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = x + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Interval for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Find points of intersection:\u003Cbr />\n- Set \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 9 - x^2\u003C/math-field>\u003C/math-field> equal to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = x + 1\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>9 - x^2 = x + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Rearrange and simplify:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2 + x - 8 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Factor the quadratic equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(x - 2)(x + 4) = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2 \\quad \\text{and} \\quad x = -4\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Valid intersection within the given interval is at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Calculate area between curves from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>:\u003Cbr />\n- Integral of top curve minus bottom curve:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int_{-1}^{2} ((9 - x^2) - (x + 1)) \\, dx\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\int_{-1}^{2} (8 - x^2 - x) \\, dx\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve the integral:\u003Cbr />\n- Integrate term-by-term:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\left[ 8x - \\frac{x^3}{3} - \\frac{x^2}{2} \\right]_{-1}^{2}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Evaluate at upper and lower limits:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left( 8(2) - \\frac{2^3}{3} - \\frac{2^2}{2} \\right) - \\left( 8(-1) - \\frac{(-1)^3}{3} - \\frac{(-1)^2}{2} \\right)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Simplify and calculate:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\left( 16 - \\frac{8}{3} - 2 \\right) - \\left( -8 + \\frac{1}{3} - \\frac{1}{2} \\right)\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 14 - \\frac{8}{3} + 8 - \\frac{1}{3} + \\frac{1}{2}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 22 - \\frac{9}{3} + \\frac{3}{6}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 22 - 3 + 0.5\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 19.5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. The area between the curves from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field> is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{19.5}\u003C/math-field>\u003C/math-field>",1091,218,"find-the-area-between-y-9-x-2-and-y-x-1-from-x-1-to-x-2",{"id":105,"category":28,"text_question":106,"photo_question":30,"text_answer":107,"step_text_answer":8,"step_photo_answer":8,"views":108,"likes":109,"slug":110},537914,"Integration by parts for \\int \\ln x dx :","Solution:\u003Cbr />\n1. Identify the components of integration by parts:\u003Cbr />\n - Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>u = \\ln x\u003C/math-field>\u003C/math-field>, then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>du = \\frac{1}{x} \\, dx\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>dv = dx\u003C/math-field>\u003C/math-field>, then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>v = x\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Apply the integration by parts formula \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int u \\, dv = uv - \\int v \\, du\u003C/math-field>\u003C/math-field>:\u003Cbr />\n * The formula becomes:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int \\ln x \\, dx = x \\ln x - \\int x \\left(\\frac{1}{x} \\right) \\, dx\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Simplify the remaining integral:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int x \\left(\\frac{1}{x} \\right) \\, dx = \\int 1 \\, dx = x + C\u003C/math-field>\u003C/math-field>, where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>C\u003C/math-field>\u003C/math-field> is the integration constant.\u003Cbr />\n\u003Cbr />\n4. Substitute back to get the final result:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int \\ln x \\, dx = x \\ln x - x + C\u003C/math-field>\u003C/math-field>.\u003Cbr />",258,52,"integration-by-parts-for-int-ln-x-dx",{"id":112,"category":28,"text_question":113,"photo_question":30,"text_answer":114,"step_text_answer":8,"step_photo_answer":8,"views":115,"likes":116,"slug":117},537913,"show how to do 349-231 in a base 12 system","Solution:\u003Cbr />\n1. Convert the numbers 349 and 231 from base 12 to decimal:\u003Cbr />\n- 349 base 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(3 \\times 12^2) + (4 \\times 12^1) + (9 \\times 12^0) = (3 \\times 144) + (4 \\times 12) + (9 \\times 1) = 432 + 48 + 9 = 489\u003C/math-field>\u003C/math-field>\u003Cbr />\n- 231 base 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(2 \\times 12^2) + (3 \\times 12^1) + (1 \\times 12^0) = (2 \\times 144) + (3 \\times 12) + (1 \\times 1) = 288 + 36 + 1 = 325\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Subtract the converted numbers in decimal:\u003Cbr />\n- Subtraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>489 - 325 = 164\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the result back to base 12:\u003Cbr />\n- The highest power of 12 in 164 is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12^1\u003C/math-field>\u003C/math-field> (since \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12^2 = 144\u003C/math-field>\u003C/math-field> is too large).\u003Cbr />\n- Divide 164 by 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>164 \\div 12 = 13 \\text{ remainder } 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n- 13 in base 12 is represented as \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 \\times 12 + 1\u003C/math-field>\u003C/math-field> remainder is 1.\u003Cbr />\n- Therefore, 164 in base 12 is 118.\u003Cbr />\n\u003Cbr />\n4. Therefore, 349 base 12 minus 231 base 12 equals: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>118\u003C/math-field>\u003C/math-field> base 12.",943,189,"show-how-to-do-349-231-in-a-base-12-system",{"id":119,"category":28,"text_question":120,"photo_question":30,"text_answer":121,"step_text_answer":8,"step_photo_answer":8,"views":122,"likes":123,"slug":124},537912,"show how to do 14+231 in a base 12 system","Solution:\u003Cbr />\n1. Express the numbers in their base 10 equivalent:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} = 1 \\cdot 12^1 + 4 \\cdot 12^0 = 12 + 4 = 16\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>231_{12} = 2 \\cdot 12^2 + 3 \\cdot 12^1 + 1 \\cdot 12^0 = 288 + 36 + 1 = 325\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Add the numbers in base 10:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>16 + 325 = 341\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the sum back to base 12:\u003Cbr />\n- To convert 341 to base 12, divide by 12 and keep track of the remainders.\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>341 \\div 12 = 28\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> (the rightmost digit)\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>28 \\div 12 = 2\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2 \\div 12 = 0\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Read the remainders from last to first to get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>245_{12}\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nThus, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} + 231_{12} = 245_{12}\u003C/math-field>\u003C/math-field>.",1182,236,"show-how-to-do-14-231-in-a-base-12-system",{"id":126,"category":28,"text_question":127,"photo_question":30,"text_answer":128,"step_text_answer":8,"step_photo_answer":8,"views":129,"likes":130,"slug":131},537911,"Task 4.(1.5 points)There are several students in the class and some pairs are friends (the friendship relationship is symmetrical). Every student has at least one friend. The whole class came to MatFyz, where each student chooses exactly one of the lectures: either mathematics or computer science. Can you prove that (in each such class) the students can divide themselves into two lectures so that each studentZhad at least one friend who attended a different lecture than the student himself Z.","1. Consider the students and friends as a graph where each student is a vertex, and an undirected edge exists between two vertices if both students are friends. \u003Cbr />\n2. The problem requires proving that this graph is bipartite. \u003Cbr />\n3. A graph is bipartite if and only if it has no odd-length cycles.\u003Cbr />\n4. Start with any vertex and perform BFS or DFS to try to color the graph using two colors, such that no two adjacent vertices share the same color. \u003Cbr />\n5. If successful, you have a bipartite graph, meaning that nodes can be divided into two groups, each corresponding to one lecture.\u003Cbr />\n6. If at any point you find a vertex with the same color during the graph traversal across an edge, there exists an odd-length cycle.\u003Cbr />\n7. The presence of such a cycle would contradict the criteria as friendship is symmetrical, ensuring each cycle detected can be colored alternately, validating bipartiteness.\u003Cbr />\n8. Since each student has at least one friend, and that ensures the graph has no isolated points, every path will connect properly into two sets.\u003Cbr />\n9. Thus, the students can always be divided into two lectures maintaining the condition.\u003Cbr />\n\u003Cbr />\nAnswer: The students can always be divided in a manner where each student has at least one friend attending a different lecture.",1006,201,"task-4-1-5-points-there-are-several-students-in-the-class-and-some-pairs-are-friends-the-friendship-relationship-is-symmetrical-every-student-has-at-least-one-friend-the-whole-class-came-to-matfy",{"id":133,"category":28,"text_question":134,"photo_question":30,"text_answer":135,"step_text_answer":8,"step_photo_answer":8,"views":136,"likes":137,"slug":138},537910,"There are 120 seats in the cinema. Eighth grades come with the same number of students. If three grades come, there will be 54 free seats left. How many seats will be left if four grades come?\n Can you solve the problem in the form of a proportion or ratio?","Ako u dvoranu dolaze tri razredna odjela i ostaje 54 slobodna mjesta, to znači da dolazi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x\u003C/math-field>\u003C/math-field> učenika, gdje je \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> broj učenika u jednom razredu.\u003Cbr />\n\u003Cbr />\nBroj zauzetih mjesta s tri razreda:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 54 = 66\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nDakle:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x = 66 \\Rightarrow x = \\frac{66}{3} = 22\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nBroj učenika u jednom razredu je \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>22\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nAko dolaze četiri razreda to znači da dolazi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4 \\times 22 = 88\u003C/math-field>\u003C/math-field> učenika.\u003Cbr />\n\u003Cbr />\nBroj slobodnih mjesta s četiri razreda:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 88 = 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nDakle, ostat će \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>32\u003C/math-field>\u003C/math-field> slobodna mjesta.",777,155,"there-are-120-seats-in-the-cinema-eighth-grades-come-with-the-same-number-of-students-if-three-grades-come-there-will-be-54-free-seats-left-how-many-seats-will-be-left-if-four-grades-come-can-yo",{"id":140,"category":28,"text_question":141,"photo_question":30,"text_answer":142,"step_text_answer":8,"step_photo_answer":8,"views":143,"likes":144,"slug":145},537909,"There are 120 seats in the cinema. Eighth grades come with the same number of students. If three grades come, there will be 54 free seats left. How many seats will be left if four grades come?","1. Izračunaj broja mjesta zauzetih za 3 razreda: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x\u003C/math-field>\u003C/math-field> \u003Cbr />\n2. Postavi jednadžbu: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 3x = 54\u003C/math-field>\u003C/math-field> \u003Cbr />\n3. Izračunaj broj učenika u jednom razredu: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x = 66\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{66}{3} = 22\u003C/math-field>\u003C/math-field> \u003Cbr />\n4. Izračunaj broj mjesta zauzetih za 4 razreda: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x = 4 \\times 22 = 88\u003C/math-field>\u003C/math-field> \u003Cbr />\n5. Izračunaj broj slobodnih mjesta za 4 razreda: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 88 = 32\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cbr />\nOdgovor: Broj učenika u jednom razredu je 22, a broj slobodnih mjesta kada dođe 4 razreda je 32.",942,188,"there-are-120-seats-in-the-cinema-eighth-grades-come-with-the-same-number-of-students-if-three-grades-come-there-will-be-54-free-seats-left-how-many-seats-will-be-left-if-four-grades-come",{"id":147,"category":28,"text_question":148,"photo_question":30,"text_answer":149,"step_text_answer":8,"step_photo_answer":8,"views":150,"likes":54,"slug":151},537908,"Determine the probability that when throwing a die, it will land:\n a) number 5\n b) a number less than 7\n c) number 1 or 4","a) The die is a cube with six faces and equal probability for each face. The number 5 appears on one face. Thus, the probability is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 \\text{ favorable outcome}}{6 \\text{ total outcomes}} = \\frac{1}{6} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) All numbers on a standard die (1 to 6) are less than 7, so all outcomes are favorable:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{6 \\text{ favorable outcomes}}{6 \\text{ total outcomes}} = 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) The numbers 1 and 4 appear, each on one face of the die:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{2 \\text{ favorable outcomes (1 or 4)}}{6 \\text{ total outcomes}} = \\frac{1}{3} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer for a): $\\frac{1}{6}$ \u003Cbr />\nAnswer for b): $1$ \u003Cbr />\nAnswer for c): $\\frac{1}{3}$",1299,"determine-the-probability-that-when-throwing-a-die-it-will-land-a-number-5-b-a-number-less-than-7-c-number-1-or-4",{"id":153,"category":28,"text_question":154,"photo_question":30,"text_answer":155,"step_text_answer":8,"step_photo_answer":8,"views":156,"likes":157,"slug":158},537907,"Ivan learned 15 out of 20 questions. What is the probability that the student got a question he did not learn?","- Ivan je naučio 15 pitanja od 20, tako da je broj pitanja koja nije naučio \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 20 - 15 = 5 \u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Ukupan broj pitanja je 20.\u003Cbr />\n- Vjerojatnost da je Ivan dobio pitanje koje nije naučio iznosi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{5}{20} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Jednostavnim deljenjem dobijamo vjerojatnost: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{5}{20} = \\frac{1}{4} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nOdgovor je: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1}{4} \u003C/math-field>\u003C/math-field>.",518,104,"ivan-learned-15-out-of-20-questions-what-is-the-probability-that-the-student-got-a-question-he-did-not-learn",{"id":160,"category":28,"text_question":161,"photo_question":30,"text_answer":162,"step_text_answer":8,"step_photo_answer":8,"views":163,"likes":164,"slug":165},537906,"561-( t+154)=137","1. Start with the equation:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>561-(t+154)=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Distribute the negative sign:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>561-t-154=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the left side by combining like terms:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>407-t=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Subtract 407 from both sides:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-t=137-407\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify the right side:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-t=-270\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Multiply both sides by -1:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t=270\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t = 270\u003C/math-field>\u003C/math-field>",1257,251,"561-t-154-137",{"first":6,"last":167,"prev":8,"next":10},181,{"current_page":6,"from":6,"last_page":167,"links":169,"path":203,"per_page":204,"to":204,"total":205},[170,173,176,178,180,182,184,187,190,193,196,199,201],{"url":6,"label":171,"active":172},"1",true,{"url":10,"label":174,"active":175},"2",false,{"url":13,"label":177,"active":175},"3",{"url":16,"label":179,"active":175},"4",{"url":19,"label":181,"active":175},"5",{"url":22,"label":183,"active":175},"6",{"url":185,"label":186,"active":175},7,"7",{"url":188,"label":189,"active":175},8,"8",{"url":191,"label":192,"active":175},9,"9",{"url":194,"label":195,"active":175},10,"10",{"url":197,"label":198,"active":175},180,"180",{"url":167,"label":200,"active":175},"181",{"url":10,"label":202,"active":175},"Next »","https://api.math-master.org/api/question",20,3601,{"data":207},[208,209,210,211,212,213],{"id":6,"title":7,"slug":8},{"id":10,"title":11,"slug":8},{"id":13,"title":14,"slug":8},{"id":16,"title":17,"slug":8},{"id":19,"title":20,"slug":8},{"id":22,"title":23,"slug":8},{"data":215},{"id":216,"category":28,"slug":217,"text_question":218,"photo_question":8,"text_answer":219,"step_text_answer":8,"step_photo_answer":8,"views":220,"likes":221,"expert":222},535056,"timber-inc-invested-profits-of-140-000-00-in-a-gic-at-4-99-compounded-semi-annually-how-long-will-it-take-for-the-investment-to-grow-to-a-value-of-176-930-82","Timber Inc. invested profits of $140,000.00 in a GIC at 4.99% compounded semi-annually. How long will it take for the investment to grow to a value of $176,930.82?","It will take approximately 4.75 years for the investment to grow to a value of $176,930.82. \n\n(Rounded to two decimal places)",395,79,{"id":223,"name":224,"photo":225,"biography":226,"created_at":8,"updated_at":8,"rating":227,"total_answer":228},35,"Hester","https://api.math-master.org/img/experts/35/35.webp","Mathematics has been my favourite subject since childhood. I studied mathematics in secondary education and secured full marks in this subject which gave me great joy and boost as i was attached to this subject. Then I opted Pre-Engineering in my higher secondary education which was mainly about the mathematics. Then I got admission in Engineering University which is ranked as Number 1 Engineering University of my country and studied different subjects of mathematics there on graduate level. I really enjoy while solving the maths questions and get too much involved in the basics and different branches of maths as it is referred as mother of sciences and I am emotionally attached to this subject.\r\n\r",4.8,111,{"data":230},{"questions":231},[232,236,240,244,248,252,256,260,264,268,272,276,280,284,288,292,296,300,304,308],{"id":233,"category":28,"text_question":234,"slug":235},532093,"Karina has a plot of 5000 square meters in which she has decided that 60% of it will be used to plant vegetables. Of this part, 12% will be dedicated to planting lettuce. How much surface area of the plot will be used to grow lettuce?","karina-has-a-plot-of-5000-square-meters-in-which-she-has-decided-that-60-of-it-will-be-used-to-plant-vegetables-of-this-part-12-will-be-dedicated-to-planting-lettuce-how-much-surface-area-of-the",{"id":237,"category":28,"text_question":238,"slug":239},533909,"Express the following numbers in decimal system,\n where the subscript indicates the base: 110101 (SUBINDEX=2)","express-the-following-numbers-in-decimal-system-where-the-subscript-indicates-the-base-110101-subindex-2",{"id":241,"category":28,"text_question":242,"slug":243},533939,"[(36,000,000)(0.000003)^2]divided(0.00000006)","36-000-000-0-000003-2-divided-0-00000006",{"id":245,"category":28,"text_question":246,"slug":247},533949,"Consider numbers from 1 to 2023. We delete 3 consecutive numbers so, that the avarage of the left numbers is a whole number","consider-numbers-from-1-to-2023-we-delete-3-consecutive-numbers-so-that-the-avarage-of-the-left-numbers-is-a-whole-number",{"id":249,"category":28,"text_question":250,"slug":251},534122,"The durability of a tire of a certain brand is a Normal random variable with an average of 64,000 km and a standard deviation of 9,000 km. Assuming independence between tires, what is the probability that the 4 tires on a car will last more than 58,000 km?","the-durability-of-a-tire-of-a-certain-brand-is-a-normal-random-variable-with-an-average-of-64-000-km-and-a-standard-deviation-of-9-000-km-assuming-independence-between-tires-what-is-the-probability",{"id":253,"category":28,"text_question":254,"slug":255},534227,"reduce the expression (7.5x 12)÷0.3","reduce-the-expression-7-5x-12-0-3",{"id":257,"category":28,"text_question":258,"slug":259},534255,"sum of 7a-4b+5c, -7a+4b-6c","sum-of-7a-4b-5c-7a-4b-6c",{"id":261,"category":28,"text_question":262,"slug":263},534308,"3+7","3-7",{"id":265,"category":28,"text_question":266,"slug":267},534313,"The question is using rule 72 determine Kari wants to save 10,000 for a down payment on a house. Illustrate the difference in years it will take her to double her current 5,000 savings based on 6%, 12% and 18% interest rate .","the-question-is-using-rule-72-determine-kari-wants-to-save-10-000-for-a-down-payment-on-a-house-illustrate-the-difference-in-years-it-will-take-her-to-double-her-current-5-000-savings-based-on-6-12",{"id":269,"category":28,"text_question":270,"slug":271},534329,"At the dance there are 150 boys the rest are girls. If 65% are girls what is the total amount in the room","at-the-dance-there-are-150-boys-the-rest-are-girls-if-65-are-girls-what-is-the-total-amount-in-the-room",{"id":273,"category":28,"text_question":274,"slug":275},534339,"X~N(2.6,1.44). find the P(X\u003C3.1)","x-n-2-6-1-44-find-the-p-x-3-1",{"id":277,"category":28,"text_question":278,"slug":279},534346,"The probability of growing a seedling from a seed is 0.62. How many seeds do I need to plant so that the probability of growing at least one seedling is greater than or equal to 0.87?","the-probability-of-growing-a-seedling-from-a-seed-is-0-62-how-many-seeds-do-i-need-to-plant-so-that-the-probability-of-growing-at-least-one-seedling-is-greater-than-or-equal-to-0-87",{"id":281,"category":28,"text_question":282,"slug":283},534347,"Buffalo Company makes and sells shampoo. Each unit requires $1.40 labor costs, material costs per unit are $0.90 and other variable costs are $0.30. It sells shampoo for $4.45 to retailers. Fixed costs are $15,000. It sold 25,000 units in the current month.\n\nWhat is the Break-Even point in units?\nWhat is the Break-Even point in dollars?\nWhat is the contribution margin of Buffalo Company?","buffalo-company-makes-and-sells-shampoo-each-unit-requires-1-40-labor-costs-material-costs-per-unit-are-0-90-and-other-variable-costs-are-0-30-it-sells-shampoo-for-4-45-to-retailers-fixed-cost",{"id":285,"category":28,"text_question":286,"slug":287},534348,"2.380× (1+0.05) / 0.95−0.05","2-380-1-0-05-0-95-0-05",{"id":289,"category":28,"text_question":290,"slug":291},534349,"The volume of a cube decreases at a rate of 10 m3/s. Find the rate at which the side of the cube changes when the side of the cube is 2 m.","the-volume-of-a-cube-decreases-at-a-rate-of-10-m3-s-find-the-rate-at-which-the-side-of-the-cube-changes-when-the-side-of-the-cube-is-2-m",{"id":293,"category":28,"text_question":294,"slug":295},534463,"Calculate the area of the parallelogram with adjacent vertices (1,4, −2), (−3,1,6) 𝑦 (1, −2,3)","calculate-the-area-of-the-parallelogram-with-adjacent-vertices-1-4-2-3-1-6-y-1-2-3",{"id":297,"category":28,"text_question":298,"slug":299},534492,"Find the equation of a straight line that has slope 3 and passes through the point of (1, 7) .\nWrite the equation of the line in general forms","find-the-equation-of-a-straight-line-that-has-slope-3-and-passes-through-the-point-of-1-7-write-the-equation-of-the-line-in-general-forms",{"id":301,"category":28,"text_question":302,"slug":303},534526,"there are 500,000 bacteria at the end of a pin point. 1000 bacteria can make a person sick. then bacteria at the tip of a pin point can make 500 people sick. Also, many people do not know that bacteria can (reproduce). Let's say there are 5 bacteria and we leave it for 15 minutes. bacteria will multiply to 10. if left for up to 30 minutes, 20 bacteria will form. if left up to 45 minutes. bacteria will multiply up to 40. every 15 minutes the bacteria will double 2. if you start with five bacteria that reproduce every 15 minutes, how manu bacteria would you have after 12 hours ?","there-are-500-000-bacteria-at-the-end-of-a-pin-point-1000-bacteria-can-make-a-person-sick-then-bacteria-at-the-tip-of-a-pin-point-can-make-500-people-sick-also-many-people-do-not-know-that-bacteri",{"id":305,"category":28,"text_question":306,"slug":307},534535,"Calculate NPV, IRR and PAYBACK through a cash flow for a period of five years, with\n discount rate of:\n a) 10%\n b) 12%\n c) 15%\n\n initial annual cost $41,400,000","calculate-npv-irr-and-payback-through-a-cash-flow-for-a-period-of-five-years-with-discount-rate-of-a-10-b-12-c-15-initial-annual-cost-41-400-000",{"id":309,"category":28,"text_question":310,"slug":311},534560,"Identify the slope and y intercept \ny=11+2/3x","identify-the-slope-and-y-intercept-y-11-2-3x",{"data":313},{"questions":314},[315,319,323,327,331,335,339,343,347,351,355,359,363,367,371,375,379,383,387,391],{"id":316,"category":28,"text_question":317,"slug":318},533921,"A food delivery company charges on average a delivery fee of $5 per order (including food and shipping) and has monthly fixed costs of $600. If the average cost of each meal delivered that is revenue for the company is $10 and the company has a monthly profit of $800, how many orders must they deliver per month?","a-food-delivery-company-charges-on-average-a-delivery-fee-of-5-per-order-including-food-and-shipping-and-has-monthly-fixed-costs-of-600-if-the-average-cost-of-each-meal-delivered-that-is-revenue",{"id":320,"category":28,"text_question":321,"slug":322},533927,"The graph of the equation x²= 4py is a parabola with focus F(_,_) and directrix y=_____ Therefore, the graph of x²=12y is a parabola with focus F(_,_) and a directrix y=_____","the-graph-of-the-equation-x-4py-is-a-parabola-with-focus-f-and-directrix-y-therefore-the-graph-of-x-12y-is-a-parabola-with-focus-f-and-a-directrix-y",{"id":324,"category":28,"text_question":325,"slug":326},534015,"(5u + 6)-(3u+2)=","5u-6-3u-2",{"id":328,"category":28,"text_question":329,"slug":330},534043,"The equation of the circle that passes through (5,3) and is tangent to the abscissa axis at x=2 is\n a.(x-2)^2 (y 3)^2 = 9\n b.(x-2)^2 (y-3)^2 = 9\n c.(x-2)^2 (y-3)^2 = 4\n d.(x-2)^2 (y 1)^2 = 4\n e.(x-2)^2 (y-1)^2 = 4","the-equation-of-the-circle-that-passes-through-5-3-and-is-tangent-to-the-abscissa-axis-at-x-2-is-a-x-2-2-y-3-2-9-b-x-2-2-y-3-2-9-c-x-2-2-y-3-2-4-d-x-2-2-y-1-2-4-e-x-2",{"id":332,"category":28,"text_question":333,"slug":334},534097,"Pedro had 80% of the amount needed to buy a game. Of this amount, you spent 15% on a watch and therefore, you will need to add another R$640.00 to purchase this game. Is the value of the game?","pedro-had-80-of-the-amount-needed-to-buy-a-game-of-this-amount-you-spent-15-on-a-watch-and-therefore-you-will-need-to-add-another-r-640-00-to-purchase-this-game-is-the-value-of-the-game",{"id":336,"category":28,"text_question":337,"slug":338},534103,"Convert 78 percent to a decimal","convert-78-percent-to-a-decimal",{"id":340,"category":28,"text_question":341,"slug":342},534117,"According to a survey in a country 27% of adults do not own a credit card suppose a simple random sample of 800 adults is obtained . Describe the sampling distribution of P hat , the sample proportion of adults who do not own a credit card","according-to-a-survey-in-a-country-27-of-adults-do-not-own-a-credit-card-suppose-a-simple-random-sample-of-800-adults-is-obtained-describe-the-sampling-distribution-of-p-hat-the-sample-proportion",{"id":344,"category":28,"text_question":345,"slug":346},534177,"The sum of two numbers is 144. Double the first number minus thrice the second number is equal to 63. Determine the first two numbers.","the-sum-of-two-numbers-is-144-double-the-first-number-minus-thrice-the-second-number-is-equal-to-63-determine-the-first-two-numbers",{"id":348,"category":28,"text_question":349,"slug":350},534312,"The sick-leave time of employees in a firm in a month is normally with a mean of 100 hours and a standard deviation of 20 hours. Find the probability that the sick-leave time of an employee in a month exceeds 130 hours.","the-sick-leave-time-of-employees-in-a-firm-in-a-month-is-normally-with-a-mean-of-100-hours-and-a-standard-deviation-of-20-hours-find-the-probability-that-the-sick-leave-time-of-an-employee-in-a-month",{"id":352,"category":28,"text_question":353,"slug":354},534324,"Solve equations by equalization method\n X-8=-2y\n 2x+y=7","solve-equations-by-equalization-method-x-8-2y-2x-y-7",{"id":356,"category":28,"text_question":357,"slug":358},534337,"If the regression equation is given by 4x –y + 5 = 0, then the slope of regression line of y on x is","if-the-regression-equation-is-given-by-4x-y-5-0-then-the-slope-of-regression-line-of-y-on-x-is",{"id":360,"category":28,"text_question":361,"slug":362},534407,"How to convert 45 kg into grams","how-to-convert-45-kg-into-grams",{"id":364,"category":28,"text_question":365,"slug":366},534415,"In a 24 hours period, the average number of boats arriving at a port is 10. Assuming that boats arrive at a random rate that is the same for all subintervals of equal length (i.e. the probability of a boat arriving during a 1 hour period the same for every 1 hour period no matter what). \r\n\r\nCalculate the probability that more than 1 boat will arrive during a 1 hour period. (P(X>1)\r\n )\r\n\r\n\r\n\r\n\r\n\r\nGive your answers to 4 decimal places and in a range between 0 and 1","in-a-24-hours-period-the-average-number-of-boats-arriving-at-a-port-is-10-assuming-that-boats-arrive-at-a-random-rate-that-is-the-same-for-all-subintervals-of-equal-length-i-e-the-probability-of-a",{"id":368,"category":28,"text_question":369,"slug":370},534450,"In poker, a full house consists of five cards, where two of the cards have the\nsame number (or letter) and the remaining three also have the same number (or\nletter) as each other (but not as the previous two cards). Use a search engine or\nWikipedia to understand the concept better if necessary.\nIn how many different ways can one obtain a full house?","in-poker-a-full-house-consists-of-five-cards-where-two-of-the-cards-have-the-same-number-or-letter-and-the-remaining-three-also-have-the-same-number-or-letter-as-each-other-but-not-as-the-previ",{"id":372,"category":28,"text_question":373,"slug":374},534457,"At the end of a lively discussion within your study group, your class neighbor, for the relevance of your points of view, asks your opinion on the subject of their debate which is the following question Am I the slave of my unconscious?\n Solve the problem posed by this subject in an argumentative production.","at-the-end-of-a-lively-discussion-within-your-study-group-your-class-neighbor-for-the-relevance-of-your-points-of-view-asks-your-opinion-on-the-subject-of-their-debate-which-is-the-following-questi",{"id":376,"category":28,"text_question":377,"slug":378},534479,"Solve for B write your answer as a fraction or as a whole number. B-1/7=4","solve-for-b-write-your-answer-as-a-fraction-or-as-a-whole-number-b-1-7-4",{"id":380,"category":28,"text_question":381,"slug":382},534510,"Consider the function f(x)=1/2(x+1)^2-3. Use the preceding/following interval method to estimate the\ninstantaneous rate of change at 𝑥 = 1.","consider-the-function-f-x-1-2-x-1-2-3-use-the-preceding-following-interval-method-to-estimate-the-instantaneous-rate-of-change-at-x-1",{"id":384,"category":28,"text_question":385,"slug":386},534522,"if y=1/w^2 yw=2-x; find dy/dx","if-y-1-w-2-yw-2-x-find-dy-dx",{"id":388,"category":28,"text_question":389,"slug":390},534585,"Marc, Jean and Michelle have traveled a lot. Marc drove twice as much as Jean, but it was Michelle who drove the most with 100km more than Marc. They respected their objective of not exceeding 1350km of distance. How far did John drive?","marc-jean-and-michelle-have-traveled-a-lot-marc-drove-twice-as-much-as-jean-but-it-was-michelle-who-drove-the-most-with-100km-more-than-marc-they-respected-their-objective-of-not-exceeding-1350km",{"id":392,"category":28,"text_question":393,"slug":394},534595,"The slope of the tangent line to the curve f(x)=4tan x at the point (π/4,4)","the-slope-of-the-tangent-line-to-the-curve-f-x-4tan-x-at-the-point-4-4",{"data":396},[397,401,405],{"id":398,"question":399,"answer":400},123750,"What is the slope of a line passing through the points (2, 5) and (-3, 9)?","The slope can be calculated using the formula: slope = (change in y)/(change in x). Plugging in the values, we get slope = (9-5)/(-3-2) = 4/-5 = -0.8. Therefore, the slope of the line is -0.8.",{"id":402,"question":403,"answer":404},135231,"What is the length of side a in a triangle with side lengths b=9, c=12, and angle A=75 degrees?","The length of side a can be found using the Sine Law. a/sin(A) = c/sin(C)a/sin(75) = 12/sin(C)a = (12 * sin(75)) / sin(C)To find sin(C), we use the fact that the sum of angles in a triangle is 180 degrees.C = 180 - A - BC = 180 - 75 - 90C = 15 degreesNow, substitute the values into the equation:a = (12 * sin(75)) / sin(15)Using a calculator, the approximate value of a is 24.56 units.",{"id":406,"question":407,"answer":408},136449,"Math question: Find the length of the hypotenuse in a right triangle if the two perpendicular sides are 3 and 4.","Answer: Using the Pythagorean theorem (a² + b² = c²), where a = 3 and b = 4, we can solve for c. Plugging in the values, we get 3² + 4² = c². Simplifying this equation gives us 9 + 16 = c². Therefore, c² = 25 and taking the square root of both sides, we find c = 5. 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