Problem 33 - Algebra 1

Question

Use the following information. From 1894 to 1903 the number of miles of cable car track decreased by about \(10 \%\) per year. There were 302 miles of track in 1894 . Sketch a graph of the results.

Step-by-Step Solution

Answer

The graph will show an exponential decay curve starting at 302 miles in 1894, and decreasing about 10% each year. The curve will show the continuous decrease in the track length over the years from 1894 to 1903. Detailed values for each year were calculated by annually reducing the previous year's distance by 10%.

Step 1: Decipher the Percentage Decay

First, determine the decay percentage, which in this case is 10%. This means that each year, the amount of cable car tracks shrinks to 90% of its value from the previous year. So, to get the next year's length of tracks, the length of tracks in the current year has to be multiplied by 0.90.

Step 2: Calculate Values for Each Year

Next, calculate the length of the tracks for each year from 1894 to 1903 using the decay percentage. Start with the length in 1894 which is 302 miles, and multiply it by 0.90 for each subsequent year. Keep the track of the year and the corresponding track length.

Step 3: Sketch the Graph

On graph paper, make a scatter plot of the year versus the track length. The horizontal axis should represent the years ranging from 1894 to 1903 and the vertical axis should represent the length of the tracks. Mark each year with its corresponding track length on the graph.

Step 4: Draw the Decay Curve

Lastly, sketch the exponential decay curve through the points in the graph. This curve will show a decreasing trend, reflecting how the length of tracks decreased over the span of 10 years.

Key Concepts

Understanding Graphing in Exponential Decay

Graphing is a powerful tool to visually represent data and observe trends over time. In exponential decay, graphing helps us illustrate how quickly a quantity decreases. For the problem with the cable car tracks, we are dealing with exponential decay because the number of miles decreases by a fixed percentage each year. To create a graph, you'll need graph paper or software that allows for plotting.

• The horizontal axis (x-axis) will represent time, spanning from 1894 to 1903.
• The vertical axis (y-axis) will represent the number of track miles.

Start by marking each year along the x-axis and the corresponding track length on the y-axis. Connect the points to form a curve, which should gradually slope downward. This line represents the exponential decay, showing that the number of track miles significantly reduced each year.

The Role of Percentage Calculation in Exponential Decay

Percentage calculation is central to understanding exponential decay. In this scenario, the number of cable car tracks decreased by 10% each year. This constant rate of decrease is what defines exponential decay. Let's break down how percentage calculations apply here. You start with the initial amount, 302 miles, and decrease it by a percentage each year. To find out how much remains each year:

• Calculate 10% of the current year's track length.
• Subtract that result from the current length.
• Alternatively, you can multiply the current length by 0.90 (which represents the remaining 90% each year).

Repeat this calculation for each year. This systematic reduction illustrates how exponential decay operates through continuous percentage calculations.

Exploring Mathematical Modeling in Exponential Decay

Mathematical modeling is a way to describe real-life situations using mathematical language and techniques, allowing us to predict outcomes based on given data. In the context of exponential decay, it enables us to forecast how a quantity changes over time.To model the cable track length over time, we use the exponential decay formula:\[ A = P(1 - r)^t \]Here:

• \(A\) is the amount remaining after time \(t\).
• \(P\) is the initial amount (302 miles).
• \(r\) is the rate of decay as a decimal (0.10 in this case).
• \(t\) is the time that has passed in years.

By substituting these values into the formula for each year, we can calculate the track length. This model helps us understand the rapid reduction in track mileage over the years due to exponential decay.