Question

similar figures and volume practice
complete this assessment to review what youve learned. it will not count toward your grade.
two similar cylinders have volumes of ( 512pi \text{ in.}^3 ), ( 64pi \text{ in.}^3 ), and ( 64pi \text{ in.}^3 ), ( 512pi \text{ in.}^3 ). by comparing volume, what is the scale factor of the small cylinder to the larger cylinder? (1 point)
the scale factor of the small cylinder to the larger cylinder is ( \frac{1}{2} )
incorrect

• scale factor is the ratio between similar figures, volume is a cubic measurement of scale factor

check answer remaining attempts 1

Explanation:

Step1: Recall the formula for the ratio of volumes of similar solids

For two similar solids, if the scale factor of the first to the second is \(k\), then the ratio of their volumes \(V_1\) and \(V_2\) is \(V_1/V_2 = k^{3}\). Let \(V_{small}=64\pi\) and \(V_{large}=512\pi\). Then \(\frac{V_{small}}{V_{large}}=\frac{64\pi}{512\pi}\).
Simplify \(\frac{64\pi}{512\pi}=\frac{64}{512}=\frac{1}{8}\).

Step2: Solve for the scale factor \(k\)

Since \(\frac{V_{small}}{V_{large}}=k^{3}\), and \(\frac{V_{small}}{V_{large}}=\frac{1}{8}\), then \(k^{3}=\frac{1}{8}\).
Take the cube - root of both sides: \(k = \sqrt[3]{\frac{1}{8}}\).
We know that \(\sqrt[3]{\frac{1}{8}}=\frac{\sqrt[3]{1}}{\sqrt[3]{8}}\) (using the property \(\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}},b
eq0\)). Since \(\sqrt[3]{1} = 1\) and \(\sqrt[3]{8}=2\), \(k=\frac{1}{2}\).

Answer:

The scale factor of the small cylinder to the larger cylinder is \(\frac{1}{2}\)