Question

a cylinder with radius ( r_1 = 3 ) and height ( h_1 = 18 ) sits inside a larger cylinder with radius ( r_2 = 5 ) and height ( h_2 ), as shown below. the two triangles within the cylinders are similar.
what is the volume, ( v_2 ), of the larger cylinder?
( v_2=\frac{1458}{125}pi )
( v_2 = 750pi )
( v_2 = 150pi )
( v_2 = 96pi )

Explanation:

Step1: Use the property of similar triangles

Since the two triangles are similar, we have the proportion \(\frac{h_1}{r_1}=\frac{h_2}{r_2}\). Given \(r_1 = 3\), \(h_1=18\), \(r_2 = 5\). Substitute the values into the proportion: \(\frac{18}{3}=\frac{h_2}{5}\).
Cross - multiply: \(3h_2=18\times5\).
Solve for \(h_2\): \(h_2=\frac{18\times5}{3}=30\).

Step2: Calculate the volume of the larger cylinder

The volume formula for a cylinder is \(V=\pi r^{2}h\). For the larger cylinder with \(r = r_2=5\) and \(h = h_2 = 30\).
Substitute into the formula: \(V_2=\pi\times5^{2}\times30\).
Calculate \(5^{2}=25\), then \(V_2=\pi\times25\times30 = 750\pi\).

Answer:

\(V_2 = 750\pi\) (the second option)