Question

a volleyball has a surface area of 67.24π square inches. what is the diameter, in inches, of the volleyball?

Explanation:

Step1: Recall the surface area formula for a sphere.

The surface area \( S \) of a sphere is given by the formula \( S = 4\pi r^2 \), where \( r \) is the radius of the sphere. We know the surface area of the volleyball (which is a sphere) is \( 67.24\pi \) square inches. So we set up the equation:
\( 4\pi r^2 = 67.24\pi \)

Step2: Solve for \( r^2 \).

Divide both sides of the equation by \( 4\pi \):
\( r^2=\frac{67.24\pi}{4\pi} \)
The \( \pi \) terms cancel out, and \( \frac{67.24}{4} = 16.81 \), so \( r^2 = 16.81 \)

Step3: Solve for \( r \).

Take the square root of both sides. Since radius is a positive quantity, we consider the positive square root:
\( r=\sqrt{16.81} \)
\( \sqrt{16.81} = 4.1 \) (because \( 4.1\times4.1 = 16.81 \))

Step4: Find the diameter.

The diameter \( d \) of a sphere is related to the radius by \( d = 2r \). Substitute \( r = 4.1 \) into this formula:
\( d = 2\times4.1 = 8.2 \)

Answer:

\( 8.2 \)