The problem asks us to find the volume of a funnel, which is in the shape of a cone. The radius of the funnel is 6 cm, and the slant height is 10 cm. We need to calculate the volume to the nearest tenth of a cubic centimeter.

1. Problem Description

The problem asks us to find the volume of a funnel, which is in the shape of a cone. The radius of the funnel is 6 cm, and the slant height is 10 cm. We need to calculate the volume to the nearest tenth of a cubic centimeter.

2. Solution Steps

First, we need to find the height $h$ of the cone. We can use the Pythagorean theorem:

$r^2 + h^2 = l^2$, where $r$ is the radius, $h$ is the height, and $l$ is the slant height.

$6^2 + h^2 = 10^2$

$36 + h^2 = 100$

$h^2 = 100 - 36$

$h^2 = 64$

$h = \sqrt{64}$

$h = 8$

Now that we have the height $h = 8$ cm and the radius $r = 6$ cm, we can find the volume of the cone using the formula:

$V = \frac{1}{3}\pi r^2 h$

$V = \frac{1}{3}\pi (6^2) (8)$

$V = \frac{1}{3}\pi (36) (8)$

$V = \frac{1}{3}\pi (288)$

$V = 96\pi$

Now we approximate the value of $\pi \approx 3.14159$:

$V \approx 96 \times 3.14159$

$V \approx 301.59264$

Rounding to the nearest tenth of a cubic centimeter, we get $301.6$ cm$^3$.

3. Final Answer

301.6 cm$^3$