A farmer wishes to paint the side of a cylindrical grain silo of height 65 feet

Question

A farmer wishes to paint the side of a cylindrical grain silo of height 65 feet and diameter 24 feet. If the paint is to be applied in a coat 1/8 inch thick, use differentials to approximate the volume of paint that the farmer needs to buy. 17 Pi ft^3 Calculate the actual error in the approximation. ft^3 What is the relative error when compared to actual volume of paint required? (Round your answer to six decimal places.) 0.022157 X Recall that the volume of a cylinder with radius r and height h is V = Pir^2h. Convert all units to feet and write the volume V as a function of r since the height of the can is constant. Then calculate the differential dV= V?(r)dr and approximate the volume of paint needed. Directly calculate the volume of paint needed and use this volume to find the errors. How many feet is 1/8 of an inch? What is dr? What is V?(r)? How is the relative error calculated from the actual error and actual volume? How much paint does the farmer need to buy?

Explanation / Answer

V = pi * r^2 * h

V = pi * r^2 * 65

dV = pi * 2r * 65 * dr

dV = 130pi * r * dr

r = 12 feet

dr = 1/8 inch = 1/96 feet

dV = 130pi * 12 * 1/96

dV = 130pi * 1/8

dV = 130pi/8

dV = 65pi/4 -----> FIRST ANSWER

Volume without paint = pi * 12^2 * 65 = 9360pi

Volume with paint = pi * (12 + 1/96)^2 * 65 = 9376.2570529513888889pi

Actual volume of paint needed = 9376.2570529513888889pi - 9360pi = 51.073038

Actual error in the approximation is : 51.073038 - 65pi/4 ----> 0.022157 ---> SECOND ANSWER

Relative error = (dV / actual volume)

RE = 0.022157 / 51.073038

RE = 0.000434 ----> THIRD ANSWER

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