Question

what is the diameter of a sphere with a volume of 89,822 in³, to the nearest tenth of an inch?

Explanation:

Step1: Recall the volume formula of 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. We know the volume \( V = 89822\space\text{in}^3 \), and we need to find the radius first and then the diameter (since diameter \( d = 2r \)).

First, we start with the volume formula:
\( 89822=\frac{4}{3}\pi r^{3} \)

Step2: Solve for \( r^{3} \)

Multiply both sides of the equation by \( \frac{3}{4\pi} \) to isolate \( r^{3} \).
\( r^{3}=\frac{3\times89822}{4\pi} \)
Calculate the numerator: \( 3\times89822 = 269466 \)
So, \( r^{3}=\frac{269466}{4\pi}=\frac{134733}{2\pi}\approx\frac{134733}{2\times3.14159}\approx\frac{134733}{6.28318}\approx21443.4 \)

Step3: Solve for \( r \)

Take the cube - root of both sides to find \( r \).
\( r=\sqrt[3]{21443.4}\approx27.8 \) (using a calculator to find the cube - root)

Step4: Find the diameter \( d \)

Since the diameter \( d = 2r \), substitute \( r\approx27.8 \) into the formula.
\( d = 2\times27.8 = 55.6 \)

Answer:

The diameter of the sphere is approximately \( 55.6 \) inches.