Question

A heavy rope, 60 ft long, weighs 0.7 lb/ft and hangs over the edge of a building 130 ft high. (let x be the distance in feet below the top of the building. enter xi* as xi.) (a) how much work w is done in pulling the rope to the top of the building? show how to approximate the required work by a riemann sum. lim n?????? n 7xi??? 10??? correct: your answer is correct. ??x i = 1 express the work as an integral. 60 correct: your answer is correct. 0 7x 10??? correct: your answer is correct. dx evaluate the integral. 42 incorrect: your answer is incorrect. ft-lb (b) how much work w is done in pulling half the rope to the top of the building? show how to approximate the required work by a riemann sum. lim n?????? n 7xi??? 10???+ changed: your submitted answer was incorrect. your current answer has not been submitted. ??x i = 1 express the work as an integral. 30 correct: your answer is correct. 0 dx evaluate the integral. ft-lb

Answer 1

Answer:

A. 1,260 lb/ft

B. 315 lb/ft

Explanation:

(a) firs we divide the rope into n sub-intervals . each sub-interval has length Δx=$\frac{60}{n}$.

next we pick any ith sub-interval, $X_{i}^*$ ∈ $[X_{i-1}, X{i}]$

the work required to move $X_{i}^*$ to the top of the building is given by

$W_{i} = F_{i}d_{i}$ = [(0.7)(Δx)][$X_{i}^*$]

 thus work required to pull the rope to the top of the building is

W = ∑ $W_{i}$ = $\lim_{n \to \infty}$∑ $0.7X_{i}^*$Δx

               = $\int\limits^{60}_0 {0.7x} \, dx = [0.7\frac{x^2}{2} ]_0^{60}$

               = 1,260 lb/ft

(B) To find the work to pull half of the rope to  the top of the building, we simply integrate using the limit 0 to 30

$W=\int\limits^{30}_0 {0.7x} \, dx = [0.7\frac{x^2}{2} ]_0^{30}$

     = 315 lb/ft