Question

which has a larger volume, a cube of sides of 8 feet or a sphere with a diameter of 8 feet? explain your reasoning.

Explanation:

Step1: Calculate volume of cube

The formula for the volume of a cube is \( V_{cube}=s^3 \), where \( s \) is the side length. Given \( s = 8 \) feet.
\( V_{cube}=8^3=8\times8\times8 = 512 \) cubic feet.

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. The diameter \( d = 8 \) feet, so the radius \( r=\frac{d}{2}=\frac{8}{2}=4 \) feet.
Substitute \( r = 4 \) into the formula: \( V_{sphere}=\frac{4}{3}\pi(4)^3=\frac{4}{3}\pi\times64=\frac{256}{3}\pi\approx\frac{256}{3}\times3.14\approx267.95 \) cubic feet.

Step3: Compare the two volumes

We have \( V_{cube}=512 \) and \( V_{sphere}\approx267.95 \). Since \( 512>267.95 \), the cube has a larger volume.

Answer:

The cube of side 8 feet has a larger volume. The volume of the cube is \( 512 \) cubic feet (calculated as \( 8^3 \)) and the volume of the sphere (with diameter 8 feet, radius 4 feet) is \( \frac{4}{3}\pi(4)^3\approx267.95 \) cubic feet. Since \( 512>267.95 \), the cube's volume is larger.