#### To determine

**To find:**

The time it will take to fill the tankunder the given conditions

#### Answer

10.97 hours

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Calculation:**

Use a vertical coordinate x measured from the water tank.

The topmost and bottommost points of the tank have coordinates x=-12 ft and x=12 ft respectively.

The radius of the tank is 12 ft

A thin horizontal slice of water is a with a width ∆x at distance x ft from the center of tank

By using Pythagoras theorem,

122=x2+radius of horizontal slice of water(disk)2

Therefore,

Radius of horizontal slice of water (disk) is 122-x2 as in the figure.

Leta thin horizontal slice of water (disk) be at distance xi* ft from the center of tank

Radius of each slice is 122-xi*2

Volume of each horizontal slice of water (disk) is π122-xi*2∆x.The total distance the slice travels to reach xi* is 72-xi*.

The total work needed to fill the tank is approximated by a Riemann sum ∑i=1n62.5 π122-xi*272-xi* ∆x

Therefore, total work is

W=limn→∞∑i=1n62.5 π122-xi*272-xi* ∆x=∫-121262.5 π122-x272-xdx

=62.5 π∫-121272122-x2-x122-x2 dx

=62.5 π2∫01272122-x2 dx the is because the second function is odd and first is even

=125π72122x-x33012

=9000π123-1233

=9000π23123

=10,368,000π ft-lb

The 1.5 horsepower pump does 1.5550=825 ft-lb of work per second.

To fill the tank, it will take

10,368,000π ft-lb825 ft-lb/s≈39,481 s≈39,4813600≈10.97 hours

**Conclusion:**

To fill the tank, it will take ≈10.97 hours