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