Question

3 select the correct answer from each drop - down menu. a conical container can hold 120π cubic centimeters of water. the diameter of the base of the container is 12 centimeters. the height of the container is centimeters. if its diameter and height were both doubled, the containers capacity would be times its original capacity.

Explanation:

1. First, find the radius \(r\) of the base of the cone:

• Given the diameter \(d = 12\) cm, then the radius \(r=\frac{d}{2}=6\) cm.
• The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\).
• We know that \(V = 120\pi\) cubic - centimeters and \(r = 6\) cm. Substitute these values into the volume formula:
• \(120\pi=\frac{1}{3}\pi(6)^{2}h\).
• First, simplify the right - hand side: \(\frac{1}{3}\pi(6)^{2}h=\frac{1}{3}\pi\times36h = 12\pi h\).
• Then, solve for \(h\):
• Set \(120\pi=12\pi h\). Divide both sides by \(12\pi\), we get \(h = 10\) cm.

1. Then, consider the situation when the diameter and height are doubled:

• The new radius \(r_{new}=2r\) and the new height \(h_{new}=2h\).
• The new volume \(V_{new}=\frac{1}{3}\pi r_{new}^{2}h_{new}=\frac{1}{3}\pi(2r)^{2}(2h)\).
• Expand the right - hand side: \(V_{new}=\frac{1}{3}\pi\times4r^{2}\times2h = 8\times\frac{1}{3}\pi r^{2}h\).
• Since \(V=\frac{1}{3}\pi r^{2}h\), the new volume is 8 times the original volume.

Answer:

(The height of the container is) 10
(If its diameter and height were both doubled, the container's capacity would be) 8