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James is trying to shoot three balls of different sizes consecutively through a hoop with a diameter of 18 inches. The balls have volumes of $850 \pi, 900 \pi$, and $950 \pi$ cubic inches. How many of the balls will fit through the hoop?

Solution

VerifiedGiven the volumes of the balls, we find their corresponding diameter length. A ball will fit through the hoop if its diameter is less than 18 inches.

The volume of a sphere (the shape of the ball) is:

$V=\dfrac{4}{3}\pi r^3$

where $r$ is the radius of the sphere.

$\textbf{For the $850\pi$ cubic inch-ball:}$

$850\pi=\dfrac{4}{3}\pi r^3$

$\dfrac{850\pi}{\dfrac {4}{3}\pi}= r^3$

$r^3=637.5$

$r=\sqrt[3]{637.5}\text{ in.}$

So, $d=2r=2\sqrt[3]{637.5}\approx 17.2\text{ in.}<18$ in. which means that this ball will fit through the hoop.

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