A water tank has the shape of an inverted right circular cone of altitude 12 ft and

Question

base radius of 6 ft. There is a hole in the bottom that is leaking at a rate of 0.3
gal/min. If water is being pumped into the tank at a rate of 10 gal/min,
a) Approximate the rate at which the water level is rising when the water is
3 ft deep. (1 gallon =0.1337 ft^2 ).
b) How long will it take for the tank to fill?

Answer 1

To solve this problem, we can use related rates. Let's first find the volume of the water in the tank as a function of the depth of the water.

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

Given that the base radius is 6 ft and the altitude is 12 ft, we can write the volume of the water as a function of the depth of the water, h, as follows:

V = (1/3)π(6^2)(12 - h)
V = π(12)(36 - h)/3
V = 4π(36 - h)

Now, let's differentiate both sides of the equation with respect to time, t:

dV/dt = d/dt (4π(36 - h))
dV/dt = -4π(dh/dt)

We know that water is being pumped into the tank at a rate of 10 gal/min, which is equivalent to 10/0.1337 = 74.83 ft^3/min.

So, dV/dt = 74.83 ft^3/min.

We also know that the water is leaking out of the hole at a rate of 0.3 gal/min, which is equivalent to 0.3/0.1337 = 2.2414 ft^3/min.

So, the net rate at which the water is increasing in the tank is given by:

dV/dt = 74.83 ft^3/min - 2.2414 ft^3/min
dV/dt = 72.5886 ft^3/min

Now, we need to find the rate at which the water level is rising when the water is 3 ft deep, which means h = 3 ft.

To find dh/dt, we can substitute the given values into the expression:

72.5886 = -4π(dh/dt)
dh/dt = -72.5886 / (4π)
dh/dt = -18.2759 / π

Now, let's approximate this value using π ≈ 3.14159:

dh/dt ≈ -18.2759 / 3.14159
dh/dt ≈ -5.8213 ft/min

a) The rate at which the water level is rising when the water is 3 ft deep is approximately -5.8213 ft/min.

b) To find out how long it will take for the tank to fill, we can set dh/dt = 0 and solve for h.

0 = -18.2759 / π
h = 0

This means that the water level will stop rising when the tank is completely filled.

Therefore, it will take infinite time for the tank to fill.