Question

an inflatable ball can be modeled as a hollow sphere. myesha measures the outer diameter of the ball to be 50 cm. if the material that makes up the ball has a thickness of 0.75 cm, find the total volume of material that makes up the ball. round your answer to the nearest hundredth if necessary. (note: diagram is not drawn to scale.)

Explanation:

Step1: Calculate the outer radius

The outer diameter \(d = 50\) cm, so the outer radius \(R=\frac{d}{2}=\frac{50}{2}=25\) cm.

Step2: Calculate the inner radius

The thickness of the material is \(0.75\) cm. So the inner radius \(r=R - 0.75=25 - 0.75 = 24.25\) cm.

Step3: Use the formula for the volume of a sphere

The volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). The volume of the material \(V_{material}=\frac{4}{3}\pi R^{3}-\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi(R^{3}-r^{3})\).
Substitute \(R = 25\) and \(r = 24.25\) into the formula:

$$
LATEXBLOCK0
$$

\(R^{3}-r^{3}=15625 - 14260.515625 = 1364.484375\)
\(V_{material}=\frac{4}{3}\pi\times1364.484375\)

$$
LATEXBLOCK1
$$

Answer:

\(5714.74\) \(cm^{3}\)