Question 3

Question

A county planner prepares the following table showing population trends in three local towns. The first column gives the name of the town. The second column gives the population as of the last census. The third column gives the estimated increase or decrease in population since that census, expressed as a percentage of the previous population.

Town Population Percentage Change
Pleasantville x –2%
Amity Center y +5%
Harmony Corners z +7%

Choose one of these towns. Use the information from the second and third columns to write two expressions showing the current estimated population of the town in terms of x, y, or z.
Add a fourth row to the table showing the population information for the town of Concord. Concord’s population as of the last census was p people;  for its percentage change, choose a negative integer not divisible by two. Write two expressions showing the current estimated population of Concord in terms of p.
Choose a number greater than 5,000 and less than 10,000 for the value of p. Based on the percentage change you chose in part b) above, calculate the estimated current population of Concord, rounding to the nearest whole number if needed. Explain why you used the expression you did to make your calculation.

Answer 1

Let's choose the town of Amity Center for this exercise.

Step 1: Write expressions for Amity Center

• The population \(y\) from the last census is subject to a percentage increase of +5%.
• We can express the current estimated population of Amity Center as:

1. Expression 1: Current population after a 5% increase: \[ y_{\text{current}} = y + 0.05y = 1.05y \]

2. Expression 2: Another way to represent it: \[ y_{\text{current}} = y(1 + 0.05) = 1.05y \]

Step 2: Add Concord row to the table

Now we will add the information for Concord.

Concord's Population Information

• Let's choose \(p = 6,500\) as the population.
• Let's say Concord experiences a percentage change of -3%.

The updated table now looks like this:

| Town | Population | Percentage Change | |-------------------|------------|-------------------| | Pleasantville | x | –2% | | Amity Center | y | +5% | | Harmony Corners | z | +7% | | Concord | 6,500 | -3% |

Step 3: Write two expressions for Concord

Based on a -3% change:

1. Expression 1: Current population after a -3% decrease: \[ p_{\text{current}} = p - 0.03p = 0.97p \]

2. Expression 2: Another way to express Concord's estimated population: \[ p_{\text{current}} = p(1 - 0.03) = 0.97p \]

Step 4: Calculate the estimated current population of Concord

Now, substituting \(p = 6,500\) into either of the expressions for Concord:

Using Expression 1: \[ p_{\text{current}} = 0.97 \times 6,500 \] Calculating that gives: \[ p_{\text{current}} = 0.97 \times 6,500 = 6,305 \]

Explanation for the calculation

I used the expression \(p_{\text{current}} = 0.97p\) because it effectively captures the -3% change in a straightforward manner, multiplying the original population by 0.97, which represents a decrease of 3%. This allows for easy calculation of the current population while reflecting the percentage change.

Summary

The estimated current population of Concord is: \[ \text{Estimated Current Population of Concord} = 6,305 \]

Answer 2

Tell me all the answers in order.

Answer 3

Sure! Here are all the answers in order:

Step 1: Expressions for Amity Center

1. Expression 1: \[ y_{\text{current}} = 1.05y \]

2. Expression 2: \[ y_{\text{current}} = 1.05y \]

Step 2: Concord's Population Information

• Population \(p\): 6,500
• Percentage Change: -3%

| Town | Population | Percentage Change | |-------------------|------------|-------------------| | Pleasantville | x | –2% | | Amity Center | y | +5% | | Harmony Corners | z | +7% | | Concord | 6,500 | -3% |

Step 3: Expressions for Concord

1. Expression 1: \[ p_{\text{current}} = 0.97p \]

2. Expression 2: \[ p_{\text{current}} = 0.97p \]

Step 4: Estimated Current Population of Concord

• Calculation: \[ p_{\text{current}} = 0.97 \times 6,500 = 6,305 \]

Explanation for the Calculation

• The expression \(p_{\text{current}} = 0.97p\) was used because it effectively captures the -3% change, allowing for straightforward multiplication to find the current population.

Final Answer

• Estimated Current Population of Concord: 6,305