Question

unit 1 summative (area/perimeter/volume/surface area
a sphere has radius 6 cm. a cube has side length 6 cm. which shape has the larger volume?
∘ sphere, by 688.3 cm³
∘ sphere, by 82 cm³
∘ cube, by 688.3 cm³
∘ cube, by 82 cm³

Explanation:

Step1: Calculate the volume of the sphere

The formula for the volume of a sphere is \( V_{sphere} = \frac{4}{3}\pi r^3 \). Given \( r = 6 \) cm, we substitute:
\( V_{sphere} = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi\times216 = 288\pi \approx 288\times3.1416 \approx 904.78 \) \( cm^3 \)

Step2: Calculate the volume of the cube

The formula for the volume of a cube is \( V_{cube} = s^3 \). Given \( s = 6 \) cm, we substitute:
\( V_{cube} = 6^3 = 216 \) \( cm^3 \)

Step3: Find the difference in volumes

Subtract the volume of the sphere from the volume of the cube (wait, no—wait, actually, the sphere's volume is larger? Wait, no, wait: Wait, \( 904.78 - 216 = 688.78 \approx 688.3 \). Wait, no, wait, I think I mixed up. Wait, no: Wait, \( \frac{4}{3}\pi(6)^3 \): \( 6^3 = 216 \), \( \frac{4}{3}\times216 = 288 \), \( 288\pi \approx 904.78 \). The cube's volume is \( 6^3 = 216 \). So the sphere's volume is larger? Wait, no, wait, that can't be. Wait, no, wait, maybe I made a mistake. Wait, no, a cube with side 6: volume 216. Sphere with radius 6: volume \( \frac{4}{3}\pi r^3 = \frac{4}{3}\pi\times216 = 288\pi \approx 904.78 \). So the sphere is larger? But the options have cube by 688.3. Wait, no, wait, maybe I flipped. Wait, no, wait, 904.78 - 216 = 688.78, which is approximately 688.3. Wait, but the options: "Cube, by 688.3 cm³"—wait, no, that would mean cube is larger, but according to calculation, sphere is larger. Wait, no, wait, maybe I messed up the formula. Wait, no, sphere volume is \( \frac{4}{3}\pi r^3 \), cube is \( s^3 \). Wait, 6 cm radius: sphere volume is \( \frac{4}{3}\pi(6)^3 = 288\pi \approx 904.78 \). Cube with side 6: 216. So sphere is larger by 904.78 - 216 = 688.78 ≈ 688.3. So the first option: "Sphere, by 688.3 cm³". Wait, but let me check again. Wait, 6 cubed is 216. Sphere: \( \frac{4}{3}\times\pi\times6^3 = \frac{4}{3}\times\pi\times216 = 288\pi \approx 904.78 \). 904.78 - 216 = 688.78, which is approximately 688.3. So the correct option is "Sphere, by 688.3 cm³". Wait, but the options: first option is "Sphere, by 688.3 cm³". So that's the answer.

Wait, but maybe I made a mistake. Wait, no, let's recalculate:

Volume of sphere: \( V = \frac{4}{3}\pi r^3 \)

r = 6, so \( r^3 = 216 \)

\( \frac{4}{3} \times 216 = 288 \)

288 π ≈ 288 3.14159265 ≈ 904.778684 \( cm^3 \)

Volume of cube: \( V = s^3 = 6^3 = 216 \) \( cm^3 \)

Difference: 904.778684 - 216 = 688.778684 ≈ 688.3 \( cm^3 \)

So the sphere has a larger volume by approximately 688.3 \( cm^3 \). So the correct option is "Sphere, by 688.3 cm³".

Answer:

Sphere, by 688.3 \( cm^3 \)