Problem 30 - Algebra 1

Question

A summer youth camp had a declining enrollment from 1995 to \(2000 .\) The enrollment in 1995 was 320 people. Each year for the next five years, the enrollment decreased by \(2 \% .\) Copy and complete the table showing the enrollment for each year. Sketch a graph of the results. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 \\ \hline \text { Enrollment } & ? & ? & ? & ? & ? & ? \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The enrollment for each year is 320 for 1995, 314 for 1996, 308 for 1997, 302 for 1998, 296 for 1999, and 290 for 2000.

Step 1: Calculate Enrollment for Each Year

We know that the enrollment in 1995 was 320 and that it decreases by 2% every year. To calculate the enrollment for a certain year, we multiply the enrollment of the previous year by 0.98 (since it is a 2% decrease). We continue this method until we reach the year 2000. For example, to calculate the enrollment for 1996, we would do: \(320 * 0.98 = 313.6\). This number is rounded to the nearest whole number as we can't have a fraction of a person.

Step 2: Complete Table

After calculating the enrollment for each year, we can fill in the table. The completed table should look like this:\[\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 \ \hline \text { Enrollment } & 320 & 314 & 308 & 302 & 296 & 290 \ \hline \end{array}\]

Step 3: Sketch Graph

To make the graph, we use the years as the x-axis and the number of enrollments as the y-axis. From the graph, we can see the trend of the summer youth camp's declining enrollment.

Key Concepts

Percent Decrease

Understanding percent decrease is crucial when analyzing situations where values are diminishing over time. In the context of our summer youth camp example, the camp experienced a 2% decrease in enrollment annually. To compute a percent decrease, we calculate the reduction in value as a fraction of the original value. For instance, if there's a 2% decline, then the multiplier for the remaining value would be 0.98 (100% - 2% = 98%, converted to a decimal for calculation purposes).

In the case of the camp, starting with 320 attendees in 1995, we calculate the subsequent years' enrollments by multiplying the previous year's enrollment by our multiplier of 0.98. This method illustrates a constant percent decrease, where the number of people leaving the camp each year changes as the base decreases, yet the percentage remains the same.

Exponential Decay

When we talk about a consistent percent decrease over a period of time, a pattern known as exponential decay emerges. This pattern occurs in many natural processes and financial contexts, such as radioactive decay or depreciation of assets. The formula to calculate the enrollment after a given number of years with exponential decay is: \[ P = P_0 \times (1 - r)^t \] where:\

• \( P_0 \) is the initial quantity (320 people for the camp in 1995)
• \( r \) is the rate of decrease (0.02 or 2% for the camp)
• \( t \) is the time that has passed (in years)

The enrollment can thus be calculated for each year, revealing the predictable nature of exponential decay, and this predictability allows organizations to plan for the future, setting strategies to combat potential losses.

Data Representation in Tables and Graphs

A table allows us to organize data efficiently, making it easier to see relationships and patterns. Upon completing our table for the youth camp enrollment, visualizing the data in a graph becomes the next step. Graphs transform numerical data into a visual format, often making the insights more accessible and understandable.

The graph for our example would have the years plotted on the x-axis and the enrollment numbers on the y-axis. The resulting line graph would show a downward, or decreasing, trend line reflecting the steady decline of camp attendees due to the 2% annual reduction. This visualization helps identify and communicate trends at a glance which can be invaluable for presentations or quick assessments of situations where numbers matter.