To find the length of the log based on 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 of the cylinder,
- \( h \) is the height (or length) of the cylinder.
First, let's find the radius of the log. Given the diameter is 32 inches, the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{32}{2} = 16 \text{ inches} \]
Next, we can substitute the known values into the volume formula. We know the volume \( V \) is \( 10,000\pi \) cubic inches, and we have found \( r \):
\[ 10,000\pi = \pi (16^2) h \]
Now, simplify the equation:
\[ 10,000\pi = \pi (256) h \]
We can divide both sides by \( \pi \):
\[ 10,000 = 256h \]
To find \( h \), we can solve for \( h \) by dividing both sides by 256:
\[ h = \frac{10,000}{256} \]
Calculating this gives:
\[ h \approx 39.0625 \]
Now, rounding this to the nearest hundredth:
\[ h \approx 39.06 \text{ inches} \]
So, the length of the log is approximately:
\[ \text{length} = 39.06 \text{ in.} \]