Answer:
0.325 ft/min
Step-by-step explanation:
To solve we must follow the following steps:
First, the volume of a cone is given by the equation:
Vc = (1/3) * Pi * (r ^ 2) * h
where r is the radius and h is the height.
they tell us that the height is equal to twice the radius, therefore
h = 2r; r = h / 2
replacing r in the volume of the cone:
Vc = (1/3) * Pi * ((h / 2) ^ 2) * h
solving we have:
Vc = (1/3) * Pi * (1/4) * (h ^ 2) * h
Vc = (1/12) * Pi * h ^ 3
They give us the change in volume with respect to time, that is, dV / dt = 50
It can also be expressed in the following way:
dV / dt = (dV / dh) * (dh / dt)
We know V, so we can derive with respect to h, we are like this:
dV / dh = 3 * (1/12) * Pi * h ^ 2
dV / dh = (1/4) * Pi * h ^ 2
we know that h is equal to 14, therefore:
dV / dh = (1/4) * Pi * 14 ^ 2
dV / dh = (1/4) * 3.14 * 196 = 153.86
Now if we replace
dV / dt = (dV / dh) * (dh / dt)
50 = 153.86 * (dh / dt)
(dh / dt) = 50/158.36
(dh / dt) = 0.3249
This means that the change in height with respect to time is 0.325 feet per minutes.
, where B is the area of the base and h is the height. We are given the volume, and the base and height in terms of x. Our formula then looks like this: 
. Simplifying the right side we have 
. Multiply both sides by 18 to get rid of the fraction and we have 
, so x = 117.575