Question

What is the volume of water in the tank when the depth of the water is 6 meters, given the function V = π/3*h^2(12 − h)?

Answer 1

To find the volume of water in the tank when the depth \( h = 6 \) meters, we use the given volume function:

\[
V = \frac{\pi}{3} h^2 (12 - h)
\]

Substituting \( h = 6 \) into the equation, we get:

\[
V = \frac{\pi}{3} (6^2) (12 - 6)
\]

Calculating \( 6^2 \):

\[
6^2 = 36
\]

Now substituting that into our equation:

\[
V = \frac{\pi}{3} (36) (12 - 6)
\]

Next, simplify \( 12 - 6 \):

\[
12 - 6 = 6
\]

Substituting back, we have:

\[
V = \frac{\pi}{3} (36) (6)
\]

Now multiply the constants:

\[
36 \times 6 = 216
\]

So the equation for the volume simplifies to:

\[
V = \frac{\pi}{3} (216)
\]

Calculating \( \frac{216}{3} \):

\[
\frac{216}{3} = 72
\]

Thus, the volume is:

\[
V = 72\pi
\]

Therefore, the volume of water in the tank when the depth is \( 6 \) meters is

\[
\boxed{72\pi} \text{ cubic meters.}
\]