You are examining a log that you find on the ground. The diameter of the log is 32 inches, but the length of the log is unknown. Find the log’s length if its volume is 10,000π in.3 . Round your answer to the nearest hundredth.

Answer 1

To find the length of the log, 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 base, and \( h \) is the height (or length) of the cylinder.

From the problem, we know:

• The volume \( V = 10,000\pi \) in³.
• The diameter of the log is 32 inches, so the radius \( r \) is half of that:

\[ r = \frac{32}{2} = 16 \text{ inches} \]

Now we can substitute these values into the volume formula:

\[ 10,000\pi = \pi (16)^2 h \]

First, we calculate \( (16)^2 \):

\[ 16^2 = 256 \]

Now the equation becomes:

\[ 10,000\pi = \pi (256) h \]

We can divide both sides by \( \pi \) (since it is non-zero):

\[ 10,000 = 256h \]

To find the height \( h \), we divide both sides by 256:

\[ h = \frac{10,000}{256} \]

Calculating the right-hand side:

\[ h = 39.0625 \]

Rounding to the nearest hundredth, the length of the log is:

\[ \boxed{39.06} \text{ inches} \]