Question

a water tank is created by revolving the graph y=1/x about the y-axis, with the bottom of the tank at y=1. the volume of the tank is given by [v(h) = the integral from 1 to h of (pi/y^2)dy] where h is the height of the water in the tank. Initially, the tank is empty, but water begins to flow into the tank at a rate of 1.5 cubic feet per minute. Determine how fast the level of the water is rising when the water is 2 feet deep.

Answer 1

In this question we obtain volume by integration between the initial point at y=1 to the height reached which is h.

The rate is 1.5 cubic feet per minute.

Given that the water has reached a height of 2 feet, we can get the volume at this height by integration.

We integrate π / y² from 1 to 2.

Integrating π / y² we have :

-π / y from y=1 to y=2

π

Substituting the values of y, we get.
π is a constant so we put it outside the bracket.


π ( - 0.5 - ( - 1) ) = 0.5π.

Let π =3.142 then volume = 3.142 × 0.5 =1.571 cubic feet.

Since the rate is 1.5 cubic feet per minute, then the time taken to reach 2 feet is:

1.571/1.5 = 1.0473minutes.