Question

a cone has a volume of ( 50pi mathrm{cm}^{3} ) and the same radius and height as a cylinder. what is the cylinders volume?
a. ( 50pi mathrm{cm}^{3} )
b. ( 100pi mathrm{cm}^{3} )
c. ( 150pi mathrm{cm}^{3} )
d. ( 75pi mathrm{cm}^{3} )

Explanation:

Step1: Recall the volume formulas

The volume formula for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\), and for a cylinder is \(V_{cylinder}=\pi r^{2}h\).

Step2: Relate the volumes

Given \(V_{cone} = 50\pi\). Since \(V_{cone}=\frac{1}{3}V_{cylinder}\) (because \(V_{cone}=\frac{1}{3}\pi r^{2}h\) and \(V_{cylinder}=\pi r^{2}h\) with same \(r\) and \(h\)).
Let \(V_{cylinder}=x\). Then \(50\pi=\frac{1}{3}x\).

Step3: Solve for \(x\)

Multiply both sides of the equation \(50\pi=\frac{1}{3}x\) by \(3\). We get \(x = 3\times50\pi=150\pi\).

Answer:

C. \(150\pi\mathrm{cm}^{3}\)