Expert: Scotto - 3/31/2008

Question
Could you please help me with this question.  I do not understand how to answer it.  I am currently studying this type of question but we have only completed examples with conical tanks.  I know that the formula for the volume of the tank is V=1/3Bh where B=area of the base and h=height

A tank has the shape of an inverted pyramid of height 50 ft whose base is a square measuring 40 ft by 40 ft.  The liquid in the tank is being drained at the rate of 8(square root)x ft^3/min, where x represents the depth of the liquid in feet.
a. Write an expression for the volume of the liquid in the tank as a function of x.
b. Let t represent the time in minutes.  Using x and t, write a differential equation that describes this situation.
c. Find a general solution to the differential equation you wrote in part (b).
d. Assume the tank is full at time t=0.  Give an expression for the depth x as a function of t.
e. How long does it take to empty the tank?

Thank you very much for your assistance.

Answer
a. Take the equation for volume and note that if B is x^2,
then h is 5x/4.  Put both of these values into the volume equation to get it in terms of x.

b. Take the derviative of the equation and set it equal to 8t.  This will give an equation of that represents the process.

c. Solve the equation you got in b.

d. Take the equation in b and solve for x in terms of t.

e. Compute the entire volume and divide by 8.

Have a great day and hope this gives you assistance.  

Thanks -- Scotto