Question

use the size guide to determine which ball has a volume of 47,688.75\\,\mathrm{cm}^3.\
ewline size guide\
ewline ball size\
ewline junior\
ewline small\
ewline medium\
ewline large\
ewline extra large\
ewline maximum ball diameter\
ewline 45\\,\mathrm{cm}\
ewline 55\\,\mathrm{cm}\
ewline 65\\,\mathrm{cm}\
ewline 75\\,\mathrm{cm}\
ewline 85\\,\mathrm{cm}

Explanation:

Step 1: Recall the volume formula for a sphere

The volume \( V \) of a sphere is given by the formula \( V=\frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the sphere (and \( r = \frac{d}{2} \), where \( d \) is the diameter).

Step 2: Solve for the radius from the volume formula

We are given \( V = 47688.75\space cm^{3} \). First, we can rearrange the volume formula to solve for \( r \):
\[

$$\begin{align*} V&=\frac{4}{3}\pi r^{3}\\ r^{3}&=\frac{3V}{4\pi}\\ r&=\sqrt[3]{\frac{3V}{4\pi}} \end{align*}$$

\]
Substitute \( V = 47688.75 \) and \( \pi\approx3.14 \) into the formula:
\[

$$\begin{align*} r^{3}&=\frac{3\times47688.75}{4\times3.14}\\ r^{3}&=\frac{143066.25}{12.56}\\ r^{3}&\approx11390.625\\ r&=\sqrt[3]{11390.625}\\ r&\approx22.5\space cm \end{align*}$$

\]

Step 3: Find the diameter

Since \( d = 2r \), then \( d=2\times22.5 = 45\space cm \)

Step 4: Match the diameter to the size guide

From the size guide, a diameter of \( 45\space cm \) corresponds to the "Junior" ball size.

Answer:

Junior