Question

The following cylinder has a volume of 800 cm^3. The diameter is 9 cm. What is the height of the cylinder? Round to the thousandths place.

Answer 1

The volume \( V \) of a cylinder can be calculated using the formula:

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

where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder,
- and \( \pi \) is a constant approximately equal to 3.14159.

Given that the volume \( V = 800 \, \text{cm}^3 \) and the diameter of the cylinder is 9 cm, we first find the radius:

\[
r = \frac{\text{diameter}}{2} = \frac{9}{2} = 4.5 \, \text{cm}
\]

Now, substituting the known values into the volume formula:

\[
800 = \pi (4.5)^2 h
\]

Calculating \( (4.5)^2 \):

\[
(4.5)^2 = 20.25
\]

Now, substituting this value into the volume equation:

\[
800 = \pi (20.25) h
\]

To isolate \( h \), we first divide both sides by \( \pi (20.25) \):

\[
h = \frac{800}{\pi \times 20.25}
\]

Calculating \( \pi \times 20.25 \):

\[
\pi \approx 3.14159 \implies \pi \times 20.25 \approx 63.617251
\]

Now, substituting this back into the equation for \( h \):

\[
h = \frac{800}{63.617251}
\]

Calculating the height \( h \):

\[
h \approx 12.582128 \, \text{cm}
\]

Rounding to the thousandths place:

\[
h \approx 12.582 \, \text{cm}
\]

Thus, the height of the cylinder is approximately:

\[
\boxed{12.582} \, \text{cm}
\]