To analyze the waffle cone with a slant height of 8 inches and a radius of 3 inches, we can find various attributes of the cone, such as its height and volume.
Step 1: Calculate the height of the cone
We can use the Pythagorean theorem to find the height (h) of the cone. The slant height (l), radius (r), and height (h) of the cone are related by the formula:
\[ l^2 = r^2 + h^2 \]
Given:
- Slant height \( l = 8 \) inches
- Radius \( r = 3 \) inches
Substituting the known values into the equation:
\[ 8^2 = 3^2 + h^2 \]
\[ 64 = 9 + h^2 \]
Now, subtract 9 from both sides:
\[ 64 - 9 = h^2 \]
\[ 55 = h^2 \]
Now, take the square root:
\[ h = \sqrt{55} \approx 7.42 \text{ inches} \]
Step 2: Calculate the volume of the cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
We now substitute \( r = 3 \) inches and \( h \approx 7.42 \) inches into the volume formula:
\[ V = \frac{1}{3} \pi (3^2) (\sqrt{55}) \]
Calculating \( 3^2 \):
\[ = \frac{1}{3} \pi (9) (\sqrt{55}) \]
Now, simplifying:
\[ V = 3 \pi \sqrt{55} \approx 3 \times 3.14159 \times 7.42 \]
\[ \approx 69.89 \text{ cubic inches} \]
Summary
- Height of the cone: \( \approx 7.42 \) inches
- Volume of the cone: \( \approx 69.89 \) cubic inches
If you have any further questions or need additional calculations, feel free to ask!
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