Question

the traditional game of tenpin bowling uses a bowling ball with a diameter of 8.5 inches. duckpin bowling is a variation of bowling that uses a smaller ball, which has \\( \frac { 1 } { 5 } \\) the volume of a tenpin bowling ball. what is the diameter of a duckpin bowling ball? round your answer to the nearest hundredth. inches

Explanation:

Step1: Find the radius of the tenpin bowling ball

The diameter of the tenpin bowling ball \(d = 8.5\) inches. The radius \(r=\frac{d}{2}=\frac{8.5}{2}=4.25\) inches.

Step2: Calculate the volume of the tenpin bowling ball

The volume formula for a sphere is \(V=\frac{4}{3}\pi r^{3}\). Substitute \(r = 4.25\) into the formula:
\(V_{tenpin}=\frac{4}{3}\pi(4.25)^{3}=\frac{4}{3}\pi\times76.765625\approx321.5575\) cubic inches.

Step3: Find the volume of the duckpin bowling ball

Since the duckpin bowling ball has \(\frac{1}{5}\) the volume of the tenpin bowling ball, \(V_{duckpin}=\frac{1}{5}\times V_{tenpin}\).
\(V_{duckpin}=\frac{1}{5}\times321.5575 = 64.3115\) cubic inches.

Step4: Calculate the radius of the duckpin bowling ball

Using the volume formula \(V=\frac{4}{3}\pi r^{3}\) for the duckpin ball (\(V = 64.3115\)), we solve for \(r\):

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Step5: Find the diameter of the duckpin bowling ball

The diameter \(d = 2r\). Substitute \(r\approx2.49\) inches, \(d\approx2\times2.49 = 4.98\) inches.

Answer:

\(4.98\)