rates of change

Question

Water is leaking out of an inverted conical tank at a rate of 14800.0 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 13.0 meters and the diameter at the top is 5.0 meters. If the water level is rising at a rate of 27.0 centimeters per minute when the height of the water is 3.0 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Answer 1

πStart by listing what you are given. Draw a diagram if it helps to set up the problem. Make sure the given information is in the correct units. (1m = 100cm) You can convert to either cm or m as long as everything has the same units.

 

Given:

 

Rate water leaking = 14,800 cm³min^-1

 

dh/dt = 27 cm·min^-1

 

h = 3 m

h = 300 cm

 

total tank height = 13 m = 1300 cm

 

tank diameter at top = 5 m

diameter = 500 cm

which means the radius, r = 250 cm

 

The tank is conical, so use the equation for volume of a cone. This relates the volume of the tank to the height.

 

V = (1/3)πr²h

 

As the tank drains/fills the radius is also changing but we only want to deal with two variables. Find an expression for r in terms of h and use substitution.

 

The tank is an inverted cone so the radius meets the height at a right angle, forming a right triangle.

 

radius at top = 250 cm

total tank height = 1300 cm

 

ratio: r/h = 250/1300 = 5/26

then

r = 5h/26

 

replace r with 5h/26 in the volume equation

 

V = (1/3)π(5h/26)²h

V = (1/3)π(25h²/676)h

V = (25/2028)πh³

 

Now that Volume is in terms of only height, take the derivative with respect to time.

 

dV/dt = (25/2028)π*(3)h² (dh/dt)

 

dV/dt = (25/676)πh² (dh/dt)

 

plug in the givens;

dh/dt = 27 cm·min^-1
h = 300 cm

 

dV/dt = (25/676)π(300cm)²*(27cm·min^-1)

 

dV/dt is the total change in volume. So since we need the rate at which water is being pumped in, we need to go one step further. Use the calculated total volume change (dV/dt) and the given rate of leaking 14,800 cm³·min^-1 to find the rate of incoming water.

 

dV/dt = (rate incoming water) - (rate leaking water)