Question

what happens to the volume of a cone if the height is tripled while the radius remains constant?
a. the volume is tripled
b. the volume is halved
c. the volume is doubled
d. the volume remains unchanged
a cone has a radius of 6 cm and a volume of 72π cm³. what is the height of the cone?
a. 4 cm
b. 8 cm
c. 2 cm
d. 6 cm

Explanation:

Step1: Volume formula of a cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\).

Step2: Analyze the first question

Let the original height be \(h_1\) and the new height be \(h_2 = 3h_1\), with radius \(r\) constant.
Original volume \(V_1=\frac{1}{3}\pi r^{2}h_1\), new volume \(V_2=\frac{1}{3}\pi r^{2}(3h_1)=3\times(\frac{1}{3}\pi r^{2}h_1)=3V_1\).

Step3: Analyze the second question

Given \(V = 72\pi\space cm^{3}\), \(r = 6\space cm\), use the formula \(V=\frac{1}{3}\pi r^{2}h\).
Substitute the values: \(72\pi=\frac{1}{3}\pi\times6^{2}\times h\).
Simplify: \(72\pi=\frac{1}{3}\pi\times36\times h\), then \(72\pi = 12\pi h\).
Solve for \(h\): \(h=\frac{72\pi}{12\pi}=6\space cm\).

Answer:

For the first question: A. The volume is tripled.
For the second question: D. 6 cm.