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 850π,900π850 \pi, 900 \pi, and 950π950 \pi cubic inches. How many of the balls will fit through the hoop?


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Given 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=43πr3V=\dfrac{4}{3}\pi r^3

where rr is the radius of the sphere.

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

850π=43πr3850\pi=\dfrac{4}{3}\pi r^3

850π43π=r3\dfrac{850\pi}{\dfrac {4}{3}\pi}= r^3


r=637.53 in.r=\sqrt[3]{637.5}\text{ in.}

So, d=2r=2637.5317.2 in.<18d=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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