To find the volume of the redwood tree trunk modeled as a right cylinder, we can use the formula for the volume \( V \) of a cylinder:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base of the cylinder and \( h \) is the height.
Step 1: Find the radius from the circumference
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Given that the circumference of the trunk is \( 16\pi \) ft, we can set up the equation:
\[ 2\pi r = 16\pi \]
To find \( r \), divide both sides by \( 2\pi \):
\[ r = \frac{16\pi}{2\pi} = 8 \text{ ft} \]
Step 2: Use the height
The height \( h \) of the trunk is given as \( 100 \) ft.
Step 3: Calculate the volume
Now that we have \( r = 8 \) ft and \( h = 100 \) ft, we can plug these values into the volume formula:
\[ V = \pi r^2 h = \pi (8)^2 (100) \]
Calculating \( r^2 \):
\[ 8^2 = 64 \]
Now substitute back into the volume formula:
\[ V = \pi (64)(100) = 6400\pi , \text{ft}^3 \]
Conclusion
The approximate volume of the redwood tree trunk is:
\[ \boxed{6400\pi , \text{ft}^3} \]
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