To find the height and volume of a cone with a given slant height and radius, we can use the following formulas:
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Height (h) can be found using the Pythagorean theorem: \[ s^2 = r^2 + h^2 \] where \( s \) is the slant height, \( r \) is the radius, and \( h \) is the height.
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Volume (V) of the cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \]
Given:
- Slant height (\( s \)) = 41 feet
- Radius (\( r \)) = 40 feet
- Use \( \pi = 3.14 \)
Step 1: Find the height
Using the Pythagorean theorem: \[ 41^2 = 40^2 + h^2 \] Calculating \( 41^2 \) and \( 40^2 \): \[ 1681 = 1600 + h^2 \]
Now, solving for \( h^2 \): \[ h^2 = 1681 - 1600 \] \[ h^2 = 81 \]
Now, take the square root to find \( h \): \[ h = \sqrt{81} = 9 \text{ feet} \]
Step 2: Find the volume
Now that we have the height, we can calculate the volume: \[ V = \frac{1}{3} \pi r^2 h \] Substituting the values: \[ V = \frac{1}{3} \cdot 3.14 \cdot (40)^2 \cdot 9 \] Calculating \( (40)^2 \): \[ (40)^2 = 1600 \] Now substitute this back: \[ V = \frac{1}{3} \cdot 3.14 \cdot 1600 \cdot 9 \]
Calculating \( 1600 \cdot 9 \): \[ 1600 \cdot 9 = 14400 \] Now substitute back into the volume equation: \[ V = \frac{1}{3} \cdot 3.14 \cdot 14400 \]
Now, calculate \( 3.14 \cdot 14400 \): \[ 3.14 \cdot 14400 = 45144 \] Finally, \[ V = \frac{1}{3} \cdot 45144 \approx 15048 \]
Final Answers:
- Height \( h = 9 \) feet
- Volume \( V \approx 15048 \) cubic feet