Question

what is the diameter of a sphere with a volume of 263 cm³, to the nearest tenth of a centimeter?

Explanation:

Step1: 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. We know the volume \( V = 263\space cm^{3}\), and we need to find the radius first and then the diameter (since diameter \( d = 2r\)).

Step2: Solve for the radius \( r \)

We start with the volume formula:
\[
263=\frac{4}{3}\pi r^{3}
\]
First, multiply both sides by \( \frac{3}{4} \) to isolate the \( \pi r^{3} \) term:
\[
r^{3}=\frac{263\times3}{4\pi}
\]
Calculate the numerator: \( 263\times3 = 789 \)
So,
\[
r^{3}=\frac{789}{4\pi}
\]
Now, calculate the value of \( \frac{789}{4\pi} \). We know that \( \pi\approx3.1416 \), so \( 4\pi\approx12.5664 \)
\[
r^{3}=\frac{789}{12.5664}\approx62.78
\]
Then, take the cube root of both sides to find \( r \):
\[
r=\sqrt[3]{62.78}\approx3.98
\]

Step3: Find the diameter \( d \)

Since the diameter \( d = 2r \), substitute \( r\approx3.98 \) into the formula:
\[
d = 2\times3.98 = 7.96\approx8.0
\]

Answer:

The diameter of the sphere is approximately \( \boldsymbol{8.0} \) centimeters.