Question

what is the approximate volume of a cone with a radius of 7 cm and height of 15 cm?
a. 1230.57 cm³
b. 769.69 cm³
c. 452.12 cm³
d. 601.44 cm³

if the radius of a cone is doubled, how does this affect the volume?
a. the volume is quartered.
b. the volume is halved.
c. the volume is unchanged.
d. the volume is quadrupled.

Explanation:

Step1: Volume formula of cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given \(r = 7\mathrm{cm}\) and \(h=15\mathrm{cm}\), substitute into the formula: \(V=\frac{1}{3}\times\pi\times7^{2}\times15\).

Step2: Calculate the volume

First calculate \(7^{2}\times15 = 49\times15=735\). Then \(V=\frac{1}{3}\times\pi\times735 = 245\pi\). Taking \(\pi\approx3.14\), we get \(V\approx245\times3.14 = 769.3\mathrm{cm}^{3}\approx769.69\mathrm{cm}^{3}\).

Step3: Analyze the effect of doubling the radius

Let the original radius be \(r\), original volume \(V_1=\frac{1}{3}\pi r^{2}h\). When the radius is doubled (\(r' = 2r\)), the new volume \(V_2=\frac{1}{3}\pi(2r)^{2}h=\frac{1}{3}\pi\times4r^{2}h = 4\times\frac{1}{3}\pi r^{2}h=4V_1\).

Answer:

For the first question: B. \(769.69\mathrm{cm}^{3}\)
For the second question: D. The volume is quadrupled.