Water is leaking out of an inverted conical tank at a rate of 9300 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 13 meters and the diameter at the top is 7 meters. If the water level is rising at a rate of 21centimeters per minute when the height of the water is 4 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Question

Water is leaking out of an inverted conical tank at a rate of 9300 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 13 meters and the diameter at the top is 7 meters. If the water level is rising at a rate of 21centimeters per minute when the height of the water is 4 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Recall that the volume of a right circular cone with height h and radius of the base r is given by

Answer 1

I noticed that oobleck answered this question for you when you posted

it a while ago this evening.
All you had to do is finish the arithmetic, after all you are in a Calculus class

Answer 2

Sorry If I interrupted. I just couldn't see and tried to repost it to get a similar question.

Answer 3

To solve this problem, we need to use the formula for the volume of a cone.

The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

First, let's determine the radius of the cone at a height of 4 meters. We can use similar triangles to find this.

Given:
Height of the cone, h = 13 meters
Diameter of the top of the cone = 7 meters
Radius of the top of the cone = diameter / 2 = 7 / 2 = 3.5 meters

Let's define:
Radius of the cone at height h, r = unknown
Height of the water level, H = 4 meters

Using similar triangles, we can set up the following proportion:

(r / h) = (3.5 / 13)

Cross-multiplying, we get:
r = (3.5 / 13) * h
r = (3.5 / 13) * 4
r ≈ 1.0769 meters

Now, we have the radius of the cone at a height of 4 meters, and we know the rate at which the water level is rising, which is 21 centimeters per minute.

To find the rate at which water is being pumped into the tank, we need to find the rate at which the volume is increasing.

Let's define:
Rate of water pumping, V' = unknown (in cubic centimeters per minute)
Rate of change of height, h' = 21 centimeters per minute
Radius of the cone at height h = 1.0769 meters

We can differentiate the volume equation with respect to time to find the rate of change of volume.

dV / dt = (1/3) * π * (2 * r * dr / dt * h + r^2 * dh/dt)

Substituting the given values:

9300 = (1/3) * π * (2 * 1.0769 * dr / dt * 4 + (1.0769)^2 * 21)

Now, we can solve for the rate of water pumping, V':

V' = (9300 - (1/3) * π * (2 * 1.0769 * dr / dt * 4 + (1.0769)^2 * 21)) / (1/3) * π

After evaluating this expression, we will find the rate at which water is being pumped into the tank in cubic centimeters per minute.