Question

the volume of a cone with a radius of 4 cm and height of 9 cm is?
a. 72π
b. 96π
c. 48π
d. 36π

what does the letter h represent in the volume formula of a cylinder?
a. diameter
b. volume
c. height
d. radius

if the radius of a cone and cylinder are both doubled, but the heights stay the same, what happens to their volumes?
a. both volumes stay the same.
b. both volumes quadruple.
c. both volumes double.
d. only the cylinders volume quadruples.

a cone and cylinder have the same radius of 2 inches. if their heights are both 6 inches, what is the difference in their volumes?
a. 24π in³
b. 16π in³
c. 32π in³
d. 8π in³

Explanation:

Step1: Volume of a cone formula

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given \(r = 4\ \text{cm}\) and \(h=9\ \text{cm}\), substitute into the formula: \(V=\frac{1}{3}\pi\times4^{2}\times9\).

Step2: Calculate the value

First, \(4^{2}=16\). Then \(\frac{1}{3}\times16\times9 = 48\). So \(V = 48\pi\ \text{cm}^3\).

Step3: Volume formula of a cylinder

The volume formula of a cylinder is \(V=\pi r^{2}h\), where \(h\) represents height.

Step4: Effect of doubling radius

For a cone \(V_{cone}=\frac{1}{3}\pi r^{2}h\), if \(r\) becomes \(2r\), \(V_{new - cone}=\frac{1}{3}\pi(2r)^{2}h=\frac{1}{3}\pi\times4r^{2}h = 4\times\frac{1}{3}\pi r^{2}h\). For a cylinder \(V_{cylinder}=\pi r^{2}h\), if \(r\) becomes \(2r\), \(V_{new - cylinder}=\pi(2r)^{2}h=4\pi r^{2}h\). So both volumes quadruple.

Step5: Volume of cone and cylinder

Volume of cylinder \(V_{cylinder}=\pi r^{2}h=\pi\times2^{2}\times6 = 24\pi\ \text{in}^3\). Volume of cone \(V_{cone}=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times2^{2}\times6 = 8\pi\ \text{in}^3\). The difference is \(24\pi-8\pi=16\pi\ \text{in}^3\).

Answer:

1. C. \(48\pi\)
2. C. height
3. B. Both volumes quadruple.
4. C. \(32\pi\ \text{in}^3\)