Problem 20 - 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 in passengers is approximately 1.67%.

Step 1: Calculate the Total Decrease

Subtract the final number of passengers from the initial number to find the total decrease. \[190,205 - 174,989 = 15,216\]Thus, the total decrease in the number of passengers over the 5-year period is 15,216.

Step 2: Calculate the Percentage Decrease

Determine the percentage decrease by dividing the total decrease by the initial value, and then multiply by 100.\[\text{Percentage Decrease} = \left( \frac{15,216}{190,205} \right) \times 100\]Calculating this gives:\[\text{Percentage Decrease} \approx 8.00\%\]

Step 3: Calculate the Annual Percentage Decrease

Since this decrease occurred over a period of 5 years, calculate the annual percentage decrease using the formula for average annual decrease:\[\text{Annual Decrease Rate} = \left(1 - \left(1 - \frac{8}{100}\right)^{\frac{1}{5}} \right) \times 100\]The calculation simplifies to:\[1 - \left( \frac{174,989}{190,205} \right)^{\frac{1}{5}}\]Using a calculator, this yields an approximate annual percentage decrease of 1.67%.

Key Concepts

Percentage Decrease Calculation

To understand the concept of percentage decrease calculation, let's break it down step by step. When we talk about a percentage decrease, we are measuring how much a value has reduced in terms of percent. In this context, it involves finding the difference between the old value and the new value, and then relating this difference back to the original value in percentage terms.

• First, identify the initial and final values. For railway passengers, these are 190,205 and 174,989 respectively.
• Calculate the total decrease by subtracting the final value from the initial value: \[ 190,205 - 174,989 = 15,216 \]
• Next, find the percentage decrease. This is done by dividing the total decrease by the initial value, then multiplying by 100 to convert it to a percentage: \[ \text{Percentage Decrease} = \left( \frac{15,216}{190,205} \right) \times 100 \]
• Once calculated, this gives a percentage decrease of approximately 8.00%.

Understanding this calculation is essential when you need to measure and communicate changes relative to an original value in percentage terms.

Exponential Decay

Exponential decay describes a process where quantities decrease at a rate proportional to their current value. This concept typically applies to contexts such as radioactive decay or depreciation in finance, but it can also be used to model situations like the decrease in the number of railway passengers over time.

• In the context of our railway passenger scenario, we want to find out how the decrease of passengers over 5 years fits an exponential model.
• The formula for exponential decay in calculating the annual rate involves using the average percentage decrease over the given period: \[ \text{Decay Rate} = \left(1 - \left(1 - \frac{8}{100}\right)^{\frac{1}{5}} \right) \times 100 \]
• Calculating this gives an annual decay rate, or percentage decrease, which in our case is roughly 1.67%.

Employing exponential decay is particularly useful here because it accounts for how change compounds over multiple periods.

Railway Passengers

The topic of railway passengers in this example illustrates how to practically apply mathematical concepts like percentage decrease and exponential decay. Analyzing passenger data can highlight trends crucial for transportation planning and policy-making.

• Over a specified period, the change in the number of passengers using a railway service can indicate shifts in transportation needs, preferences, or external factors like economic conditions.
• By evaluating such data, we can determine the effectiveness of transportation systems, the impact of regional developments, or the need for infrastructural improvements.
• This kind of computational analysis helps transport authorities make informed decisions, whether to allocate resources more efficiently or improve service quality.

In summary, understanding the fluctuation of railway passenger numbers using percentage and exponential models supports strategic planning and improved public transport services.