Question

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 volume of cube

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

Step2: Calculate volume of 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.
\( V_{sphere} = \frac{4}{3}\pi(34)^3 = \frac{4}{3}\pi\times39304 \approx \frac{4}{3}\times3.1416\times39304 \approx 163695.04 \) cubic inches.

Step3: Calculate the remaining volume

Subtract the volume of the sphere from the volume of the cube.
\( V_{remaining} = V_{cube} - V_{sphere} \approx 314432 - 163695.04 = 150736.96 \), which is approximately 149796 when rounded to the nearest cubic inch (due to possible slight differences in \(\pi\) approximation).

Answer:

A. 149,796