Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Amelia and Beulah are city dwellers who have set up pulley systems to get their groceries delivered without walking the stairs. Amelia pulls her basket filled with 12 pounds of groceries up to her 40 -foot-high balcony. Beulah pulls her basket filled with 16 pounds of cleaning supplies up to her 30 -foot-high window. Assuming both women use ropes weighing \(0.2 \mathrm{lb} / \mathrm{ft}\), whose task requires more work? How much more work? (Assume friction is negligible.)

Step-by-Step Solution

Answer

Amelia's task requires more work, specifically 140 foot-pounds more work than Beulah's.

Step 1: Calculate the weight of the rope for each woman

First, the weight of the rope for each woman must be calculated, as this adds to the total weight that they have to pull up. The weight of the rope can be calculated by multiplying its length (which is equal to the height of their dwelling) by the weight per foot. \n For Amelia, the weight of the rope = 40 feet * 0.2 lb/ft = 8 pounds. \n For Beulah, the weight of the rope = 30 feet * 0.2 lb/ft = 6 pounds.

Step 2: Calculate the total weight that each woman has to pull up

Next, we need to sum up the weights of the items and the rope that each woman has to pull up. \n For Amelia, the total weight = weight of groceries + weight of the rope = 12 pounds + 8 pounds = 20 pounds. \n For Beulah, the total weight = weight of cleaning supplies + weight of the rope = 16 pounds + 6 pounds = 22 pounds.

Step 3: Calculate the work done by each woman

The work done by each woman is the product of the total weight to be lifted and the height they need to be lifted to. \n For Amelia, the work done = total weight * height = 20 pounds * 40 feet = 800 foot-pounds. \n For Beulah, the work done = total weight * height = 22 pounds * 30 feet = 660 foot-pounds.

Step 4: Compare the work done by each woman

Now that we have calculated the work done by each woman, the last step is to compare their values. Amelia has done 800 foot-pounds of work while Beulah has done 660 foot-pounds of work. Therefore, Amelia has done more work.

Step 5: Calculate the difference in the work done

The difference between the work done by Amelia and Beulah can be found by subtracting the work done by Beulah from the work done by Amelia. Difference = 800 foot-pounds - 660 foot-pounds = 140 foot-pounds.

Key Concepts

Understanding Pulley Systems

Pulley systems are simple machines that make lifting tasks easier. They use a wheel with a rope or cable to change the direction of the force. While a single pulley doesn’t reduce the amount of force needed, it helps in lifting because you can apply force in a more convenient direction. In many practical situations, like Amelia and Beulah’s, pulleys allow them to lift items from a lower level to a higher one without walking the stairs. This system reduces physical strain and can be found in various scenarios, from construction to home uses.

Weight Calculation Essentials

When calculating the weight in physics problems involving pulleys, it's important to consider all elements involved. This means accounting for both the items being lifted and any additional weight like the rope. For Amelia, we calculated the weight of the groceries and the rope.

• Groceries: 12 pounds
• Rope (40 feet at 0.2 lb/ft): 8 pounds

The total weight is 20 pounds. For Beulah, the cleaning supplies and rope are calculated as follows:

• Cleaning supplies: 16 pounds
• Rope (30 feet at 0.2 lb/ft): 6 pounds

The total weight is 22 pounds. Accurately calculating these weights ensures correct work calculations.

Calculating Work Done

Work done in physics is calculated as the product of force and distance, where the force is often the weight in these scenarios. The formula is:\[\text{Work} = \text{Weight} \times \text{Height}\]For Amelia, the work done can be found by multiplying her total weight (20 pounds) by the height (40 feet), resulting in 800 foot-pounds.
For Beulah, her total work is calculated with a weight of 22 pounds at a height of 30 feet, totaling 660 foot-pounds. The concept of work done helps us understand the energy required to move objects in various situations.

Effective Mathematical Problem-Solving

Tackling physics problems often requires a systematic approach to solving concepts like work and energy. Here's how to effectively approach these problems:

• Identify known and unknown values: Start by determining which quantities are given, such as weights and heights.
• Apply relevant formulas: Use basic equations like the work formula to connect physical concepts mathematically.
• Check unit consistency: Ensure all units match throughout the calculations to avoid errors.
• Verify calculations: Double-check calculations to confirm results reflect physical reality.

By following these steps, complicated problems become more manageable, and it becomes easier to verify that your answers are correct.