Question

10 mark for review a cube has an edge length of 68 inches. a solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. to the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere? a 149,796 b 164,500 c 190,955 d 310,800

Explanation:

Step1: Calculate the volume of the cube

The formula for the volume of a cube is \( V_{cube} = s^3 \), where \( s \) is the edge length. Given \( s = 68 \) inches, so \( V_{cube} = 68^3 \).
\( 68^3 = 68\times68\times68 = 314432 \) cubic inches.

Step2: Calculate the volume of the sphere

The formula for the volume of a sphere is \( V_{sphere} = \frac{4}{3}\pi r^3 \), where \( r \) is the radius. Given \( r = 34 \) inches, so \( V_{sphere} = \frac{4}{3}\pi(34)^3 \).
First, calculate \( 34^3 = 34\times34\times34 = 39304 \). Then, \( V_{sphere} = \frac{4}{3}\pi\times39304 \approx \frac{4}{3}\times3.1416\times39304 \approx 164636 \) cubic inches (rounded to the nearest whole number).

Step3: Calculate the volume of the space not taken up by the sphere

Subtract the volume of the sphere from the volume of the cube: \( V = V_{cube} - V_{sphere} \).
\( V = 314432 - 164636 = 149796 \) cubic inches.

Answer:

A. 149,796