Calculus II PLTL Fall 2014 Worksheet 3 These problems are to be done without the use of a calculator unless otherwise specified 1 Round Robin A uniform cable hanging over the edge of a tall building is 100 ft long and weighs 175 pounds Use the following steps to compute the amount of work required 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 0 at the top or bottom of the cable To set up the integral to compute the work required to lift the whole cable you first need to find a formula for the work required to lift one small section of cable Since you are looking at a very small section of cable you can assume that the force is constant so the work to lift the section is W F d This means that you need to compute how much force is needed to pull the bit of cable and how far that bit of cable needs to be pulled Once you 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 that this section of rope will have to be lifted in terms of x c What is the force needed to move this section Compute the weight of the cross 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 to the top of the building then evaluate it 2 Scribe A bucket that weighs 70 pounds when filled with water is lifted at a constant rate by a mechanical winch from the bottom of a well that is 60 feet deep a Compute the work required to lift the bucket from the bottom of the well to the top b The chain that is being used to lift the bucket weighs 0 55 pounds per foot Find the amount of work required to lift the bucket and chain from the bottom of the well to the top c When the bucket reaches the top of the well you discover that it only weighs 35 pounds and you find a hole in the bucket Assuming that the water has been leaking out at a constant rate compute the amount of work required to lift the bucket and chain from the bottom of the well to the top 3 Round Robin Let f x x4 a 1 2 32 x graphed below on the interval 12 1 Find the arc length of the graph of f on the interval 21 1 b Set up an integral that could be used to calculate the area of the surface obtained when 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 substitution u R 1 2 2 write the integral 0 2 1 t 1 t 4 dt as an integral with respect to u 1 t 1 t to a Use the substitution to find the upper and lower bounds of the new integral say a and b respectively Rb b There were two options given that had the same bounds One was a u2 du and R b 1 u the other was a 1 u 2 du Which is correct and why 5 Pairs A rectangular storage tank for rainwater has a base of 12 feet by 20 feet and a height of 10 feet Suppose the top of the tank is located 5 feet below ground level and the tank is full of rainwater weighing 60 lb ft3 How much work will it take to pump all of the water to ground level
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