Question

1. which has a larger volume: a cylinder with a radius of 5 cm and a height of 12 cm, or a sphere with a radius of 5 cm?

a. cylinder by 188.4 cm³
b. sphere by 62.8 cm³
c. both have the same volume
d. cylinder by 418.9 cm³

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \(V_{cylinder}=\pi r^{2}h\).
Given \(r = 5\ \text{cm}\) and \(h=12\ \text{cm}\), then \(V_{cylinder}=\pi\times5^{2}\times12=\pi\times25\times 12 = 300\pi\ \text{cm}^3\).
Using \(\pi\approx3.14\), \(V_{cylinder}\approx300\times3.14 = 942\ \text{cm}^3\).

Step2: Calculate the volume of the sphere

The formula for the volume of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\).
Given \(r = 5\ \text{cm}\), then \(V_{sphere}=\frac{4}{3}\pi\times5^{3}=\frac{4}{3}\pi\times125=\frac{500}{3}\pi\ \text{cm}^3\).
Using \(\pi\approx3.14\), \(V_{sphere}\approx\frac{500}{3}\times3.14\approx523.3\ \text{cm}^3\).

Step3: Find the difference in volumes

\(V_{cylinder}-V_{sphere}=942 - 523.3=418.7\approx418.9\ \text{cm}^3\) (There is a calculation - rounding difference. If we calculate more accurately:
\(V_{cylinder}=300\pi\), \(V_{sphere}=\frac{500}{3}\pi\), \(V_{cylinder}-V_{sphere}=\pi(300-\frac{500}{3})=\pi\times\frac{900 - 500}{3}=\frac{400}{3}\pi\).
\(\frac{400}{3}\times3.14=\frac{1256}{3}\approx418.7\). If we use more precise \(\pi\) value \(\pi = 3.14159\), \(\frac{400}{3}\times3.14159\approx418.9\))

Answer:

A. Cylinder by \(188.4\ \text{cm}^3\)