A cable that weighs 8 lb/ft is used to lift 650 lb of coal up a mine shaft 600 ft deep. Find the work done. Show how to approximate the required work by a Riemann sum. (Let x be the distance in feet below the top of the shaft. Enter xi * as xi.)

Question

Mathematics

Jolly

3 March, 02:12

A cable that weighs 8 lb/ft is used to lift 650 lb of coal up a mine shaft 600 ft deep. Find the work done. Show how to approximate the required work by a Riemann sum. (Let x be the distance in feet below the top of the shaft. Enter xi * as xi.)

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Jamar Drake · Answer 1

1830000 ft*lb

Step-by-step explaination:

Work done = fd

f=force:

d = distance

f = 650 + 8x lbs, x is current amount of cable in shaft

dW = fdx = (650+8x) dx

x ranges from 0 to 600 then the total work is:

W = integral (x=0 to x=600) ((650+8x) dx)

W = (650x+4x^2) [x=0, x=600]

W = (650*600 + 4 * (600) ^2) = 1830000 ft*lb

W = 1830000 ft*lb

Riemann sum approximation:

W = sum (i=1 to i=n) (650+8xi*) (xi - x (i-1))

x0 is 0, xn is 600 (or x0 is 600, xn is 0)

assuming the partition is a constant interval deltax sum can be written as:

W = summation (i=1 to i=n) (650+8xi*) deltax