Problem 25 - Applied Calculus

Question

The number of passengers using a railway fell from 190,205 to 174,989 during a 5 -year period. Find the annual percentage decrease over this period.

Step-by-Step Solution

Answer

The annual percentage decrease is approximately 1.6%.

Step 1: Understand the Decrease

First, determine the total decrease in the number of passengers over the 5-year period. This is calculated by subtracting the final number of passengers from the initial number of passengers: \[ 190,205 - 174,989 = 15,216 \] Thus, the number of passengers decreased by 15,216 over 5 years.

Step 2: Calculate Annual Decrease

Next, calculate the average annual decrease in passengers. Since the decrease of 15,216 occurred over 5 years, we divide by 5 to find the annual decrease: \[ \frac{15,216}{5} = 3,043.2 \] Thus, the average annual decrease in the number of passengers is 3,043.2.

Step 3: Determine Initial Value

Identify the initial number of passengers at the beginning of the 5-year period, which is 190,205.

Step 4: Calculate Percentage Decrease

Calculate the annual percentage decrease using the formula: \[ \text{Percentage Decrease} = \left( \frac{\text{Annual Decrease}}{\text{Initial Value}} \right) \times 100 \] Substitute the annual decrease and the initial number of passengers:\[ \left( \frac{3,043.2}{190,205} \right) \times 100 \approx 1.6\% \] The annual percentage decrease over this period is approximately 1.6%.

Key Concepts

Annual Percentage Change Made Simple

Understanding annual percentage change is crucial when analyzing changes over a period of time, like stock prices, population, or, in this case, the number of passengers using a railway. Annual percentage change lets you see how much something increases or decreases each year on average. Imagine it as the yearly progress of a journey, broken into bite-sized, more comprehensible portions.

To find annual percentage change, follow these steps:

• Identify the total change over the period. For example, find how much the number of passengers has changed over the years.
• Calculate the annual change. This means dividing the total change by the number of years.
• Use the annual change to find the percentage. This involves comparing the annual change to the starting value, then multiplying by 100 to get it in percentage form.

In the given exercise, the annual percentage decrease is approximately 1.6%. This tells you that annually, passengers decrease by around 1.6%, making it a valuable indicator for long-term planning and adjustments.

Solving Calculus Problems Without Stress

Calculus isn't only about complicated derivatives and integrations; it helps us understand how things change over time. In real-world problems, like this railway passenger decrease, calculus can play a hidden role. While this problem doesn’t directly use calculus, understanding the principles can shed light on how such problems evolve over time.

Converting the total five-year decrease to an annual rate is a way of understanding a function's slope in simpler terms. If we had continuous data, calculus steps in by allowing us to describe this change continuously over time, not just in rounded steps.

For students just starting with calculus, think of it as a toolkit:

• Derivatives help us find how fast something is changing at any moment.
• Integrals let us sum up changes to determine a total amount.

In many cases, these tools can simplify complex problems, making it easier to calculate not just annual changes but many other real-world applications as well.

Breaking Down Arithmetic Operations

Arithmetic operations form the foundation of most mathematical concepts. They involve basic computations like addition, subtraction, multiplication, and division. Understanding these operations is essential to tackle more complex problems.

In our scenario, the arithmetic operations help track the changes in the number of passengers. Here's how they are used:

• Subtraction (\(190,205 - 174,989\)) determines the total decrease in passengers.
• Division helps distribute this total change evenly across the specified period (five years in this case) to find the annual decrease.
• Multiplication is used when converting the fractional annual decrease into a percentage, by multiplying by 100.

Mastering these operations enables you to dissect a problem into manageable steps, making sure calculations are accurate and logical. When applied correctly, they bolster your problem-solving skills across various mathematical domains.