This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. A heavy rope, 50 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 140 ft high. Approximate the required work by a Riemann sum, then express the work as an integral and evaluate it. (a) How much work W is done in pulling the rope to the top of the building?(b) How much work W is done in pulling half the rope to the top of the building?

Riemann sum

W = lim n→∞ Σ 0.4xᵢΔx (with the summation done from i = 1 to n)

Workdone in pulling the entire rope to the top of the building = 180 lb.ft

Work done in pulling half the rope to the top of the building = 135 lb.ft

Step-by-step explanation:

Using Riemann sum which is an estimation of area under a curve

The portion of the rope below the top of the building from x to (x+Δx) ft is Δx.

Then workdone in lifting this portion through a length xᵢ ft would be 0.4xᵢΔx

So, the Riemann sum for this total work done would be

W = lim n→∞ Σ 0.4xᵢΔx (with the summation done from i = 1 to n)

The Riemann sum can easily be translated to integral form.

In integral form, with the rope being 30 ft long, we have

W = ∫³⁰₀ 0.4x dx

W = [0.2x²]³⁰₀ = 0.2 (30²) = 180 lb.ft

b) When half the rope is pulled to the top of the building, 30 ft is pulled up until the length remaining is 15 ft

W = ∫³⁰₁₅ 0.4x dx

W = [0.2x²]³⁰₁₅ = 0.2 (30² - 15²) = 135 lb.ft

Hope this Helps!!!

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