A cylinder of radius 3 cm is carved out of a cylinder with a radius of 7 cm. If the two cylinders have the same height, find the volume of the remaining solid.

Steps to Solve

1. Identify the dimensions of both cylinders

The outer cylinder has a radius of (7 , \text{cm}) and height (14 , \text{cm}).
The inner cylinder (which is carved out) has a radius of (3 , \text{cm}) and the same height of (14 , \text{cm}).

2. Calculate the volume of the outer cylinder

The formula for the volume (V) of a cylinder is given by:

$$ V = \pi r^2 h $$

Using the outer cylinder's dimensions:

$$ V_{\text{outer}} = \pi (7)^2 (14) $$

Calculating:

$$ V_{\text{outer}} = \pi (49)(14) = 686\pi , \text{cm}^3 $$

3. Calculate the volume of the inner cylinder

Using the formula for the volume of a cylinder again:

$$ V_{\text{inner}} = \pi (3)^2 (14) $$

Calculating:

$$ V_{\text{inner}} = \pi (9)(14) = 126\pi , \text{cm}^3 $$

4. Calculate the volume of the remaining solid

To find the volume of the remaining solid, subtract the volume of the inner cylinder from the volume of the outer cylinder:

$$ V_{\text{remaining}} = V_{\text{outer}} - V_{\text{inner}} $$

Substituting the volumes:

$$ V_{\text{remaining}} = 686\pi - 126\pi = 560\pi , \text{cm}^3 $$