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224 CHAPTER 9. A FEW STANDARD APPLICATIONS

and then press evaluate numerically #=? on the toolbar. The result is∫ 5

0sin(x)exp

(−x2)dx = 0.42444

Actually, this software is built on Mupad which is a part of the symbolic math package ofMATLAB.

9.5 Exercises1. The main span of the Portage Lake lift bridge2 weighs 4,400,000 pounds. How much

work is done in raising this main span to a height of 100 feet?

2. A cylindrical storage tank having radius 20 feet and length 40 feet is filled with afluid which weighs 50 pounds per cubic foot. This tank is lying on its side on theground. Find the total force acting on the ends of the tank by the fluid.

3. Suppose the tank in Problem 2 is filled to a depth of 8 feet. Find an integral for thework needed to pump the fluid to a height of 50 feet.

4. A conical hole is filled with water which has weight 62.5 pounds per cubic feet. Ifthe depth of the hole is 20 feet and the radius of the hole is 10 feet, how much workis needed to pump the water to a height of 10 feet above the ground?

5. Suppose the spring constant is 2 pounds per foot. Find the work needed to stretchthe spring 3 feet beyond equilibrium.

6. A 20 foot chain lies on the ground. It weighs 5 pounds per foot. How much work isdone to lift one end of the chain to a height of 20 feet?

7. A 200 foot chain dangles from the top of a tall building. How much work is neededto haul it to the top of the building if it weighs 1 pound per foot?

8. A dam 500 feet high has a width at depth y equal to 4000−2y feet. What is the totalforce on the dam if it is filled?

9. ∗When the bucket is filled with water it weighs 30 pounds and when empty it weighs2 pounds and the person on top of a 100 foot building exerts a constant force of 40pounds. The bucket is full at the bottom but leaks at the rate of .1 cubic feet persecond. How much work does the person on the top of the building do in lifting thebucket to the top? Will the bucket be empty when it reaches the top? You can useNewton’s law that force equals mass times acceleration. You can neglect the weightof the rope.

10. In the situation of the above problem, suppose the person on the top maintains aconstant velocity of 1 foot per second and the bucket leaks at the rate of.1 pound persecond. How much work does he do and is the bucket empty when it reaches thetop?

2This is the heaviest lift bridge in the world. It joins the towns of Houghton and Hancock in the upper peninsulaof Michigan spanning Portage lake. It provides 250 feet of clear channel for ships and can provide as much as100 feet of vertical clearance. The lifting machinery is at the top of two massive towers 180 feet above the water.Aided by 1,100 ton counter weights on each tower, sixteen foot gears pull on 42 cables to raise the bridge. Thisusually creates impressive traffic jams on either side of the lake. The motion up and down of this span is quiteslow.