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A baseball has a circumference of 9 inches.  A softball has a circumference of 12 inches.  What is the approximate difference in volume between the two balls?  Round your answer to the nearest hundredths place.

May 1, 2018

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A baseball has a circumference of 9 inches.
A softball has a circumference of 12 inches.
What is the approximate difference in volume between the two balls?

$$\text{Let r_b =  radius baseball }\\ \text{Let r_s =  radius softball }$$

1. baseball

$$9 = 2\pi r_b \\ r_b = \dfrac{9}{2\pi} \\ V_b = \dfrac{4}{3}\pi r_b^3$$

2. softball

$$12 = 2\pi r_s \\ r_s = \dfrac{12}{2\pi} \\ V_s = \dfrac{4}{3}\pi r_s^3$$

3. difference in volume

$$\begin{array}{|rcll|} \hline V_s - V_b &=& \dfrac{4}{3}\pi r_s^3 - \dfrac{4}{3}\pi r_b^3 \\\\ &=& \dfrac{4}{3}\pi\left( r_s^3 - r_b^3 \right) \\\\ &=& \dfrac{4}{3}\pi\left[ \left(\dfrac{12}{2\pi} \right)^3 - \left(\dfrac{9}{2\pi} \right)^3 \right] \\\\ &=& \dfrac{4\pi}{3\cdot 8\pi^3} \left( 12^3 - 9^3 \right) \\\\ &=& \dfrac{1}{3\cdot 2\pi^2} \cdot 999 \\\\ &=& \dfrac{166.5}{\pi^2} \\\\ &=& 16.8699770764 \\ \hline \end{array}$$

$$\text{The approximate difference in volume between the two balls is \mathbf{16.87\ in^3} }$$

May 2, 2018