Question

3. Inside a cubic box with edge 35 cm, a ball has been introduced that is tangent to the faces of the cube as shown in the following figure. Calculate the volume included by the free space in the cube-shaped box.

104

likes518 views

Santino

4.5

73 Answers

Let's start by finding the radius of the ball. Since the ball is tangent to the faces of the cube, the diameter of the ball is equal to the edge length of the cube. Therefore, the radius of the ball is half of this length:

Letr be the radius of the ball. We know that the diameter of the ball is equal to the edge length of the cube, which is 35 cm. Therefore, 2r = 35 , and r = \frac{35}{2} = 17.5 cm.

Next, we need to calculate the volume of the ball:

The volume of a ball is given by the formulaV = \frac{4}{3} \pi r^3 .

Substitute the radiusr = 17.5 cm into the formula:

V=\frac{4}{3}\pi(17.5)^3\approx\frac{4}{3}\times3.1416\times17.5^3\approx4.1889\times17.5^3\approx22449.85 cubic cm.

The volume of the free space in the cube-shaped box is equal to the difference between the volume of the cube and the volume of the ball:

The volume of the cube is given byV_{\text{cube}} = s^3 , where s is the edge length of the cube.

Substitute the edge lengths = 35 cm into the formula:

V_{\text{cube}} = 35^3 = 42875 cubic cm.

Therefore, the volume included by the free space in the cube-shaped box is:

V_{\text{free space}}=V_{\text{cube}}-V_{\text{ball}}=42875-22449.35=20425.65 cubic cm.

\boxed{20425.65\text{ cubic cm}} is the volume included by the free space in the cube-shaped box.

Let

Next, we need to calculate the volume of the ball:

The volume of a ball is given by the formula

Substitute the radius

The volume of the free space in the cube-shaped box is equal to the difference between the volume of the cube and the volume of the ball:

The volume of the cube is given by

Substitute the edge length

Therefore, the volume included by the free space in the cube-shaped box is:

Frequently asked questions (FAQs)

Math Question: What is the measure of the third angle of an isosceles triangle if the two congruent angles each measure 40 degrees?

+

What is the equation for the perpendicular bisectors of a triangle?

+

What is the formula for finding the area of a triangle given the base length and height?

+

New questions in Mathematics