Calculus II PLTLFall 2014Worksheet 3These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Round Robin) A uniform cable hanging over the edge of a tall building is 100 ftlong and weighs 175 pounds. Use the following steps to compute the amount of workrequired to pull the cable to the top of the building.(a) Sketch the building and cable. Next to the cable, draw a vertical axis with x = 0at the top (or bottom) of the cable.To set up the i ntegral to compute the work required to lift the whole cable, you first needto find a formula for the work required to lift one small section of cable. Since you arelooking at a very small section of cable, you can assume that the force is constant, so thework to lift the section is W = F d. This means that you need to compute how much forceis needed to pull the bit of cable and how far that bit of cable needs to be pulled. Onceyo u have a formula for both quantities, you can set up the integral.(b) Sketch a small cross-section of the rope of width 4x. What is the distance thatthis section of rope wil l have to be lifted, in terms of x?(c) What is the force needed to move this section? Compute the weight of thecross-section and use the fact that weight is a force.(d) What is the work required to pull just the cross-section to the top of the building?(e) Set up the integral to calculate the work required to pull all 100 feet of rope tothe top of the building, then evaluate it.2) (Scribe) A bu cket that wei ghs 70 pounds wh en filled with water is lifted at a constantrate, by a mechanical winch, from the bottom of a well that i s 60 feet deep.(a) Compute the work requir ed to lift the bucket from the bottom of the well to thetop.(b) The chain that is being used to l ift the bucket weighs 0.55 poun ds per foot. Findthe amount of work required to lift the bucket and chain from the bottom of the well tothe top.(c) When the bucket reaches the top of the well, you discover that it o nly weighs 35pounds, and you find a hole in the bucket. Assuming that the water has been l eaking outat a constant rate, compute the amount of work required to lift the bu cket and chain fromthe bottom of the well to the top.3) (Round Robin) Let f(x) = x4+132x−2, graphed below on the interval [12, 1] .(a) Find the arc length of the graph of f on the interval [12, 1] .(b) Set up an integral that could be used to calculate the area of the surface obtainedwhen the portion of the graph of f described in (b) is rotated about the x-axis.4) (Scribe) On Exam 1, Problem 8 , you were asked to use the su bstitution u =1+t1−ttowrite the integralR1/202(1+t)2(1−t)4dt as an integral with respect to u.(a) Use the substitution to find the upp er and lower bounds of the new integral, saya and b, respectively.(b) There were two options given that had the same bounds. One wasRbau2du andthe other wasRba1+u(1−u)2du. Which is corr ect, and why?5) (Pairs) A rectangular storage tank for rainwater has a base of 12 feet by 20 feet anda hei g ht of 10 feet. Suppose the top of the tank is lo cated 5 feet below ground level, andthe tank is full of rainwater weighing 60 lb/ft3. How much work will it take to pu mp allof the water to ground
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