We are asked to find the volume of a sphere with a radius of $9.1$ cm. We should not round any intermediate steps and use the $\pi$ button on a calculator. We need to round the final answer to the nearest hundredth.

1. Problem Description

We are asked to find the volume of a sphere with a radius of $9.1$ cm. We should not round any intermediate steps and use the $\pi$ button on a calculator. We need to round the final answer to the nearest hundredth.

2. Solution Steps

The formula for the volume of a sphere is:

$V = \frac{4}{3} \pi r^3$

where $V$ is the volume and $r$ is the radius.

We are given that the radius $r = 9.1$ cm. Plugging this value into the formula, we get:

$V = \frac{4}{3} \pi (9.1)^3$

$V = \frac{4}{3} \pi (753.571)$

Using a calculator with the $\pi$ button, we get:

$V \approx \frac{4}{3} (3.1415926535...) (753.571)$

$V \approx \frac{4}{3} (2367.316862...)$

$V \approx 3156.422483...$

Rounding the answer to the nearest hundredth, we have:

$V \approx 3156.42$ cm$^3$

3. Final Answer

The approximate volume is $3156.42$ cm$^3$.