Question

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 10,000 pi square inches. round your answer to the nearest hundredth.

Answer 1

To find the length of the log given its volume and diameter, we can use the formula for the volume of a cylinder, which is given by:

\[
V = \pi r^2 h
\]

where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height (or length, in the case of a log).

Given that the diameter of the log is 32 inches, we can find the radius:

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

We know the volume of the log is \( 10,000 \pi \) cubic inches. We can set up the equation with the volume formula:

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

Now, we can simplify this equation by cancelling out \( \pi \) from both sides:

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

Calculating \( (16)^2 \):

\[
(16)^2 = 256
\]

So the equation now is:

\[
10,000 = 256h
\]

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

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

Calculating \( h \):

\[
h \approx 39.0625
\]

Rounding this to the nearest hundredth gives:

\[
h \approx 39.06
\]

Thus, the length of the log is approximately **39.06 inches**.