To find the height and volume of the cone, we can use the following formulas.
- Finding the Height of the Cone: The relationship between the slant height (l), height (h), and radius (r) of the cone can be expressed using the Pythagorean theorem: \[ l^2 = h^2 + r^2 \]
Given:
- Slant height \( l = 41 \) feet
- Radius \( r = 40 \) feet
We can rearrange the formula to solve for the height \( h \): \[ h^2 = l^2 - r^2 \] \[ h^2 = 41^2 - 40^2 \] \[ h^2 = 1681 - 1600 \] \[ h^2 = 81 \] \[ h = \sqrt{81} = 9 \text{ feet} \]
- Finding the Volume of the Cone: The volume \( V \) of a cone can be calculated using the formula: \[ V = \frac{1}{3} \pi r^2 h \]
Using \( \pi \approx 3.14 \), the radius \( r = 40 \) feet, and the height \( h = 9 \) feet: \[ V = \frac{1}{3} \times 3.14 \times (40)^2 \times 9 \] \[ V = \frac{1}{3} \times 3.14 \times 1600 \times 9 \] \[ V = \frac{1}{3} \times 3.14 \times 14400 \] \[ V = \frac{1}{3} \times 45216 \] \[ V = 15072 \text{ cubic feet} \]
Final Answers:
- The height of the cone is 9 feet.
- The volume of the cone is 15072 cubic feet.