### Question 1
**a) Write two expressions in terms of t showing Conor's total cost.**
1. Total cost = \( t + 0.15t \)
2. Total cost = \( 1.15t \)
**b) Choose a value for t greater than $12 and less than $13.**
Let's choose \( t = \$12.50 \).
**c) Use one of the expressions you wrote in part a) to find Conor’s total cost, rounded to the nearest penny. Show your work.**
Using \( 1.15t \):
\[
\text{Total cost} = 1.15 \times 12.50 = 14.375
\]
Rounding to the nearest penny:
\[
\text{Total cost} \approx \$14.38
\]
**d) Suppose that Conor’s total cost to see the movie is greater than $17.50 and less than $20. What could be a value of t to make that possible?**
We need to find \( t \) such that \( 17.50 < 1.15t < 20.00 \).
Solving for \( t \):
\[
\frac{17.50}{1.15} < t < \frac{20.00}{1.15}
\]
\[
15.22 < t < 17.39
\]
Therefore, \( t \) could be any value between \$15.22 and \$17.39. For example, \( t = \$16.00 \).
### Question 2
**a) Write two expressions in terms of d to show how much Winona and Takoda would pay in total if they bought their mother the set of power tools.**
1. Total cost = \( d - 0.15d \)
2. Total cost = \( 0.85d \)
**b) What does the expression 0.90(d – 5) refer to? How do you know? How else could this expression be written?**
The expression \( 0.90(d - 5) \) refers to the total cost after a 10% discount on an item that is \( d - 5 \) dollars. This can be written as:
\[
0.90(d - 5) = d - 5 - 0.10(d - 5)
\]
**c) Choose a value for d that is greater than $25 and less than $100.**
Let's choose \( d = \$50 \).
**d) Calculate the total cost of each of the possible presents for the value you chose.**
- Power tools = \( 0.85 \times 50 = \$42.50 \)
- Water skis = \( 0.75 \times (50 + 15) = 0.75 \times 65 = \$48.75 \)
- Jewelry = \( 0.90 \times (50 - 15) = 0.90 \times 35 = \$31.50 \)
**e) Write a short paragraph telling Winona and Takoda which present they should buy and explaining why.**
Given that the jewelry is the least expensive option at \$31.50, followed by the power tools at \$42.50 and the water skis at \$48.75, Winona and Takoda should consider their budget and the utility of each gift. If they want to save money and still provide a meaningful present, the jewelry would be the best choice. However, if their mother has a particular need or interest in power tools or water skis, they should weigh the cost against the utility accordingly.
### Question 3
**a) 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.**
Let's choose Pleasantville.
1. Current population = \( x - 0.02x \)
2. Current population = \( 0.98x \)
**b) Add a fourth row to the table showing the population information for the town of Concord.**
| Town | Population | Percentage Change |
|----------------|------------|-------------------|
| Pleasantville | x | -2% |
| Amity Center | y | +5% |
| Harmony Corners | z | +7% |
| Concord | p | -3% |
**c) Write two expressions showing the current estimated population of Concord in terms of p.**
1. Current population = \( p - 0.03p \)
2. Current population = \( 0.97p \)
**d) Choose a number greater than 5,000 and less than 10,000 for the value of p.**
Let's choose \( p = 8000 \).
**e) Based on the percentage change you chose in part b), calculate the estimated current population of Concord, rounding to the nearest whole number if needed.**
Using \( 0.97p \):
\[
\text{Current population} = 0.97 \times 8000 = 7760
\]
Concord's current estimated population is 7,760. Using the expression \( 0.97p \) simplifies the calculation process by applying the percentage change directly.
### Question 4
**a) Write two different expressions giving the perimeter of the donkey’s pen.**
1. Perimeter = \( 2a + 2b \)
2. Perimeter = \( 2b + 2(13b) \)
**b) Choose a value for a that is greater than 5 and less than 15.**
Let's choose \( a = 10 \).
**c) Evaluate the perimeter of the pen for that value, using one of the expressions that you wrote.**
Since \( a = 13b \) and we chose \( a = 10 \):
\[
10 = 13b \implies b = \frac{10}{13}
\]
Using \( 2a + 2b \):
\[
\text{Perimeter} = 2(10) + 2\left(\frac{10}{13}\right) = 20 + \frac{20}{13} \approx 20 + 1.54 = 21.54 \text{ yards}
\]
I used the expression \( 2a + 2b \) for clarity and simplicity in evaluating the perimeter directly with separate values for \( a \) and \( b \).
Use the worksheet to record your answers to the Algebraic Expressions portfolio activities. When you are finished, save your worksheet with your answers and submit it for a portfolio grade. Question 1 A ticket to a movie costs t dollars, and there is a 15% amusement tax on each ticket. Conor buys a ticket to see a movie. Write two expressions in terms of t showing his total cost. Choose a value for t greater than $12 and less than $13. Use one of the expressions you wrote in part a) to find Conor’s total cost, rounded to the nearest penny. Show your work. Suppose that Conor’s total cost to see the movie is greater than $17.50 and less than $20. What could be a value of t to make that possible? Question 2 Winona and Takoda are shopping for a present for their mother. The chart shows several possible presents. There is no sales tax in the state where they live. Present Cost ($) Discount (%) Total Paid ($) Power tools d dollars 15% Water skis d + 15 dollars 25% Jewelry d – 15 dollars 10% Write two expressions in terms of d to show how much Winona and Takoda would pay in total if they bought their mother the set of power tools. What does the expression 0.90(d – 5) refer to? How do you know? How else could this expression be written? Choose a value for d that is greater than $25 and less than $100. Calculate the total cost of each of the possible presents for the value you chose. Then write a short paragraph telling Winona and Takoda which present they should buy and explaining why. Question 3 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. Question 4 A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side measuring b yards and the adjacent side measuring a yards. A’Leila knows that a=(13)b . Write two different expressions giving the perimeter of the donkey’s pen. Choose a value for a that is greater than 5 and less than 15. Evaluate the perimeter of the pen for that value, using one of the expressions that you wrote. Explain why you used that expression to evaluate the perimeter of the donkey’s pen.
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