Problem 10 - Thomas Calculus

Question

Leaky sandbag A bag of sand originally weighing 144 lb was lifted at a constant rate. As it ruse, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft. How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Step-by-Step Solution

Answer

The work done lifting the sandbag to 18 ft is 1944 ft-lb.

Step 1: Understand the Problem

We need to calculate the total work done in lifting the bag of sand, which loses half its weight over 18 ft. Initially, the bag weighs 144 lb and drops to 72 lb by the time it reaches 18 ft. The sand leaks at a constant rate.

Step 2: Calculate the Rate of Sand Loss

Initially, the bag weighs 144 lb and is 72 lb when it reaches 18 ft. The sand loss rate can be expressed as \( \,r = \frac{144 \text{ lb} - 72 \text{ lb}}{18 \text{ ft}} = 4 \, \text{lb/ft}\). This means the weight per foot during the lifting decreases by 4 lb.

Step 3: Determine the Weight Function

Let \( x \) be the distance in feet. The weight of the sandbag at height \( x \) is \( w(x) = 144 - 4x \). This function accounts for the constant rate of sand loss as the bag is lifted.

Step 4: Set Up the Work Integral

The work done is given by the integral \( W = \int_{0}^{18} w(x) \, dx = \int_{0}^{18} (144 - 4x) \, dx \). This integral calculates the total work done lifting the sandbag under varying weight.

Step 5: Solve the Integral

Calculate \( W = \int (144 - 4x) \, dx = \left[ 144x - \frac{4x^2}{2} \right]_{0}^{18} \). Evaluating this gives \( W = (144 \times 18) - 2(18)^2 = 2592 - 648 = 1944 \).

Step 6: Result Interpretation

The amount of work done lifting the sandbag to 18 ft is 1944 ft-lb. This calculation assumes that the weight reduction is perfectly linear with height.

Key Concepts

Calculus Integral

In this exercise, the concept of the calculus integral is foundational for determining the total work done in lifting a sandbag while sand steadily leaks out at a constant rate. Essentially, the integral is a powerful mathematical tool that allows us to calculate the total accumulation of work as the sandbag's weight changes during the lift.

Work has a direct relationship with force and distance, and because the force exerted changes with height as sand leaks out, we use integration to account for this continuously changing force over a certain distance. Here, our integral is set up as follows:

• We need to integrate the weight function over the distance from the initial height (0 ft) to the final height (18 ft).

Thus, the integral becomes \[W = \int_{0}^{18} (144 - 4x) \, dx\]This integral gives us a way to sum up all the infinitesimal contributions of work done at each infinitesimal height layer, all the way through the full 18 ft of lift.

Variable Force

In this problem, the concept of variable force is key to understanding the nuances of lifting the leaky sandbag. While at first glance, a variable force might sound complex, it simply means that the force acting on the bag is not constant but depends on another factor—in this case, the height.

As the sandbag is lifted, sand leaks out, reducing its weight. This means that the force required to lift it changes from the initial to the final weight.

• Initially, the bag weighs 144 lb.
• At 18 ft, it weighs 72 lb due to continual sand loss.

To capture this change, we express the weight of the bag at any height, \(x\), with the function \(w(x) = 144 - 4x\). This decline in weight produces a corresponding decline in the required lifting force, creating a **variable** force scenario that our integral calculation must account for in computing the total work.

Weight Function

Creating an effective weight function allows us to quantify how the bag's weight decreases as it is lifted due to sand leakage. A function essentially represents a mathematical relationship between quantities—in this case, height and weight for the bag of sand.

The weight function is expressed as \(w(x) = 144 - 4x\), where:

• \(144\) is the initial total weight of the sandbag in pounds.
• \(4x\) represents the rate at which the sand's weight decreases per foot.
• \(x\) stands for the height in feet.

This function elegantly consolidates the problem's details into one expression that easily fits into our integral setup. As the height \(x\) increases, the weight \(w(x)\) linearly decreases, reflecting the steady leakage of sand at the rate of 4 lb per foot.

Physics Application

This exercise nicely illustrates a practical application of physics concepts, particularly in the branch of mechanics involving work and energy. When lifting objects, the work done is a product of force and distance; however, here the situation is nuanced due to the changing weight of the sandbag.

Through this realistic scenario:

• We see how integration is applied to solve physics problems involving non-uniform forces.
• We learn how leaking mass modifies the typical calculations of work.
• We apply the fundamental principle that work can be determined even when force varies with distance.

By considering real-life factors, like sand leakage, we appreciate how calculus and physics intersect to resolve complex issues, exemplified by calculating the work done in lifting the sandbag, yielding a result of 1944 ft-lb."}]}]}