Problem 47 - Applied Calculus

Question

Which relative change is bigger in magnitude? Justify your answer. Find the relative change of a population if it changes (a) From 1000 to \(2000 \quad\) (b) From 2000 to 1000 (c) From 1,000,000 to 1,001,000

Step-by-Step Solution

Answer

Part (a) has the largest relative change magnitude.

Step 1: Understand Relative Change

Relative change is the change in a quantity relative to its original value. It is computed as \( \frac{\text{new value} - \text{old value}}{\text{old value}} \).

Step 2: Calculate Relative Change for (a)

For part (a), the initial population is 1000 and the new population is 2000. Compute the relative change using the formula: \[ \text{Relative change} = \frac{2000 - 1000}{1000} = \frac{1000}{1000} = 1 \]

Step 3: Calculate Relative Change for (b)

For part (b), the initial population is 2000 and the new population is 1000. Compute the relative change: \[ \text{Relative change} = \frac{1000 - 2000}{2000} = \frac{-1000}{2000} = -0.5 \]

Step 4: Calculate Relative Change for (c)

For part (c), the initial population is 1,000,000 and the new population is 1,001,000. Compute the relative change: \[ \text{Relative change} = \frac{1,001,000 - 1,000,000}{1,000,000} = \frac{1000}{1,000,000} = 0.001 \]

Step 5: Compare the Relative Changes

Compare the magnitudes of the relative changes found: - Part (a): 1 - Part (b): 0.5 - Part (c): 0.001 The biggest magnitude of relative change is in part (a), which is 1.

Key Concepts

Population Change

Population change refers to the variation in the number of individuals residing in a specific area over a period of time. It is typically expressed using relative or absolute metrics. In this exercise, we focus on relative population change, which provides an insightful way to measure how much a population has grown or shrunk relative to its original size.

The formula to calculate relative population change is as follows:

• Identify the old or original population value.
• Identify the new or changed population value.
• Use the formula: \(\text{Relative change} = \frac{\text{new value} - \text{old value}}{\text{old value}}\)

This formula gives you a sense of proportion by scaling the change according to the initial size. It allows you to compare changes across different populations, irrespective of their original sizes.

Magnitude Comparison

When comparing changes, especially in the context of changes expressed in percentages or fractions, it is crucial to understand the magnitude of change. Magnitude refers to the absolute value of the relative change, which means how large the change is, regardless of whether it is an increase or decrease. This is important because:

• It shows the impact level of the change without the influence of direction (increase or decrease).
• Assists in recognizing which change had a greater effect relative to its initial value.

In the exercise, there are three cases of population change to compare. The crucial part here is not just noting whether it's an increase or a decrease, but rather how significant that change is relative to the initial population size. For example, in part (a), a change from 1000 to 2000 results in the highest magnitude of 1, signifying a 100% increase, whereas in part (b), the change from 2000 to 1000 is expressed as -0.5, a decrease but lesser magnitude. Understanding the magnitude helps in making sense of the relative importance or impact of different changes.

Step-by-Step Solution

Solving problems using a step-by-step approach helps greatly in breaking down complex concepts into manageable parts. Let's dive into the steps used in calculating and comparing relative changes in populations in our exercise:

1. **Understand Relative Change**: First, grasp what relative change means and how it helps to compare the extent of change relative to an initial state. 2. **Calculate for Each Case**: Using the provided formula, calculate the relative change in each scenario.

• For (a), a twofold increase from 1000 to 2000 results in a relative change of 1.
• For (b), a halving from 2000 to 1000 results in a relative change of -0.5.
• For (c), a modest increase from 1,000,000 to 1,001,000 results in a relative change of 0.001.

3. **Compare the Results**: After calculating, compare the absolute values of these results to determine their magnitude. In this solution, the biggest magnitude of relative change was observed in case (a), which was a complete doubling of the population.

This kind of methodical approach not only solves the problem but also enhances understanding, helping you avoid pitfalls or mistakes commonly associated with rushed calculations.