Question

Two 10 meter poles are 30 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used?

145

likes
723 views

Answer to a math question Two 10 meter poles are 30 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used?

Expert avatar
Hester
4.8
116 Answers
1. Define the positions of the poles and the point where the wire is staked.
2. Let the poles be positioned at points A(0, 10) and B(30, 10) respectively.
3. Let the point where the wire is staked be at a point P(x, 0).

4. The distances from the top of each pole to the staking point P are:

PA = \sqrt{x^2 + 10^2} = \sqrt{x^2 + 100}
PB = \sqrt{(30 - x)^2 + 10^2} = \sqrt{(30 - x)^2 + 100}

6. Sum these distances to get the total length of the wire as a function of x:

L(x) = \sqrt{x^2 + 100} + \sqrt{(30 - x)^2 + 100}

7. Differentiate \(L(x)\) with respect to \(x\) and set the derivative equal to zero to find the minimum:

\frac{dL}{dx} = \frac{x}{\sqrt{x^2 + 100}} + \frac{-(30 - x)}{\sqrt{(30 - x)^2 + 100}} = 0

8. Simplify to find the critical points:

\frac{x}{\sqrt{x^2 + 100}} = \frac{30 - x}{\sqrt{(30 - x)^2 + 100}}
x \sqrt{(30 - x)^2 + 100} = (30 - x) \sqrt{x^2 + 100}

9. Square both sides to eliminate the square roots and solve for x:

x^2 (30 - x)^2 + 100x^2 = (30 - x)^2 x^2 + 100(30 - x)^2
100x^2 = 100(30 - x)^2
x^2 = (30 - x)^2

10. Solving this:

x = 30 - x
2x = 30
x = 15

Therefore, the wire should be staked x = 15 meters from either pole to minimize the total length of the wire.

Frequently asked questions (FAQs)
What is the measure of angle x in a circle with a central angle of 120°?
+
What is the horizontal shift of the graph of f(x) = √(x + 4) - 2?
+
What is the limit of (3x^2 + 4x - 5) as x approaches 2?
+
New questions in Mathematics
58+861-87
4x567
(5u + 6)-(3u+2)=
(2x+5)^3+(x-3)(x+3)
A National Solidarity Bond offers A 5 year bond offering a gross return of 15% Calculate the AER for this investment. (Give your answer to two decimal places, no need for the percent or € sign in your answer)
If f(x,y)=6xy^2+3y^3 find (∫3,-2) f(x,y)dx.
A person borrows rm 1000 from a bank at an interest rate of 10%. After some time, he pays the bank rm 1900 as full and final settlement of the loan. Estimate the duration of his loan.
Convert 78 percent to a decimal
A vaccine has a 90% probability of being effective in preventing a certain disease. The probability of getting the disease if a person is not vaccinated is 50%. In a certain geographic region, 60% of the people get vaccinated. If a person is selected at random from this region, find the probability that he or she will contract the disease. (4 Points)
4+168×10³×d1+36×10³×d2=-12 -10+36×10³×d1+72×10³×d2=0
Congratulations, you have saved well and are ready to begin your retirement. If you have $1,750,000.00 saved for your retirement and want it to last for 40 years, and will earn 10.8% compounded monthly: What is the amount of the monthly distribuion? 216.50 How much interest is earned in retirement?
A company made 150,000 in the first year 145,000 in the second 140,000 in the third year successively during the first decade of this company's existence it made a total of
Write the detailed definition of a supply chain/logistics related maximization problem with 8 variables and 6 constraints. Each constraint should have at least 6 variables. Each constraint should have At least 5 variables will have a value greater than zero in the resulting solution. Variables may have decimal values. Type of equations is less than equal. Numbers and types of variables and constraints are important and strict. Model the problem and verify that is feasible, bounded and have at least 5 variables are nonzero.
You buy a $475,000 house and put 15% down. If you take a 20 year amortization and the rate is 2.34%, what would the monthly payment be?
A property sold for $745,000 in a co-brokered transaction. The seller has agreed to pay a 7% commission to the listing firm. The listing firm has agreed to equally split the commission with the selling firm. If the buyer’s broker will receive 8% of the selling firm’s commission, how much commission will the buyer’s broker receive? $14,900 $3725 $$37250 $18625
X^X =49 X=?
Gender and communication : Answer the question ( 1 paragraph is ok) . Please can you write about women? Compared to your other identities, how much of a role does gender play in your life? And has your own sex/gender offered you privileges or disadvantages? How so?
2x-4=8
Triangle ABC has AB=AC and angle BAC =X, with X being less than 60 degrees. Point D lies on AB such that CB = CD Point E lies on AC such that CE= DE Determine angle DEC in terms of X
Slope (7,3) and (9,5)