Question

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?

Answer 1

To calculate the work done in pulling the rope to the top of the building, we approximate the required work using a Riemann sum and then express it as an integral. We then evaluate the integral for the given values to determine the exact amount of work done.

Step-by-step explanation:

To calculate the work done in pulling the rope to the top of the building, we need to approximate the required work using a Riemann sum. The work done by each small segment of the rope can be calculated as the product of the weight of the segment and the distance it is lifted. We can approximate the rope as a series of small segments, each with a length of Δx. The work done by each segment can be expressed as ΔW = (weight per unit length) * Δx * (height of the building - height of the segment). By summing up the work done by all the segments, we can obtain an approximation of the total work. Now, let's express the work as an integral. Since the rope is continuous, we can take the limit as Δx approaches zero, resulting in the integral form:
W = ∫[(weight per unit length) * (height of the building - height of the segment)] dx, where x ranges from 0 to 50. Solving this integral gives us the exact value of the work done in pulling the rope to the top of the building. To evaluate this integral, we substitute the given values into the integral expression and calculate the result.

Answer 2

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!!!