Answer:
0.0063 ft / min
Step-by-step explanation:
From the problem statement we have that the variation in volume with respect to time is equal 10, like this:
dV / dt = 10 ft ^ 3 / min
We know that the volume of the cone is given by the equation:
V = (1/3) * Pi * r ^ 2 * h
Now, we are mentioned that the diameter of the base of the cone is approximately three times the altitude, therefore:
2r = 3h, solving for r we have:
r = (3/2) * h, replacing the volume equation:
V = (1/3) * Pi * [(3h / 2) ^ 2] * h, solving we have:
V = (3/4) * Pi * h ^ 3
They ask us to calculate the change in height with respect to time, that is dh / dt
Therefore we derive both sides with respect to time, we have to:
dV / dt = (3/4) * Pi [3 * (h ^ 2) dh / dt] = (9/4) * pi * (h ^ 2) * dh / dt
reorganizing for dh / dt, we have:
dh / dt = [4/9 * pi * (h ^ 2)] * dV / dt
Knowing that h is 15 and dV / dt is 10, we replace these values:
dh / dt = [4/9 * 3.14 * (15 ^ 2)] * 10 = 0.0063 ft / min
0.0063 ft / min would be the change in height with respect to time.
