Question

A heavy rope, 60 ft long, weighs 0.3 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.)
a. How much work W is done in pulling the rope to the top of the building?
b. Express the work as an integral.
c. Evaluate the integral.
d. How much work W is done in pulling half the rope to the top of the building?

Answer 1

Answer:

half the rop to the top of the building = W = 423.75

top of the building = W = 423.75

Step-by-step explanation:

a) How much work W is done in pulling the rope to the top of the building?

When a length of rope xi has been pulled to the top of the building, then the force  required to move the remaining rope is 1/2(g(60− xi)), the weight of the rope.

The work to pull  an additional small length of rope, ∆x, to the top of the building is 1 /2 ((60 − xi)∆x).

Note that  the relationship between xi and ∆x is that for n constant lengths of rope equal to ∆x, then  xi = i∆x. The Riemann sum for the work is:

W=  ∑ n,i = 1  (1/2(60 - xi). ∆x) = ∑ n,i = 1  (1/2(60 - i∆x). ∆x)

Note that xi = i∆x will not limit to zero as n → ∞. The corresponding integral is:

W =  $\int\limits^3_0 {0. 1/2 (60 - x)} \, dx$

W = $1/2. [60x - 1/2. x^2]^3_0$

W = 423.75

d) How much work W is done in pulling half the rope to the top of the building?

This requires almost the same Riemann sum except the upper limit on the sum  changes:

W=  ∑ n/2,i = 1  (1/2(60 - xi). ∆x) = ∑ n,i = 1  (1/2(60 - i∆x). ∆x)

The corresponding integral is:

W = $\int\limits^{1.5}_0 {5. 1/2 (60 - x)} \, dx$

W = $1/2. [60x - 1/2. x^2]^{1.5}_0 . 5$

W = 423.75