To find the length of the log given its volume, we can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height (length of the log).
Given:
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The diameter of the log is 32 inches, so the radius \( r \) is half of that: \[ r = \frac{32}{2} = 16 \text{ inches} \]
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The volume \( V \) is given as 10,000 cubic inches.
Rearranging the formula for height \( h \) gives us:
\[ h = \frac{V}{\pi r^2} \]
Now, substituting the values into the equation:
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Calculate \( r^2 \): \[ r^2 = 16^2 = 256 \text{ square inches} \]
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Calculate the volume with the radius: \[ h = \frac{10000}{\pi \times 256} \]
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Using \( \pi \approx 3.14159265359 \): \[ h = \frac{10000}{3.14159265359 \times 256} \]
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Calculate \( \pi \times 256 \): \[ \pi \times 256 \approx 804.247 \]
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Now calculate \( h \): \[ h \approx \frac{10000}{804.247} \approx 12.43 \text{ inches} \]
Thus, the length of the log is approximately 12.43 inches.