Question 1
Scenario Description: Let's consider a scenario where a company sells custom T-shirts at a price of $20 each. If they sell 100 T-shirts in a week, their revenue would be constant at $20,000, as each T-shirt contributes equally to revenue.
Why it Involves a Constant Rate: This scenario involves a constant rate because for every T-shirt sold (x), the revenue (y) increases by a fixed amount ($20), leading to a direct proportionality.
Mathematical Representation: This scenario can be represented as: \[ y = 20x \] where \( y \) is the total revenue and \( x \) is the number of T-shirts sold.
Question 2
Constant Ratio Explanation: In the first week, after selling 20 T-shirts, if the friends predict that they will sell 20% more each subsequent week, it forms a geometric progression so that each week involves selling a constant ratio more than the last.
Mathematical Equation Representation: The total T-shirts sold after the first week can be represented with the equation: \[ T = 20(1.2^n) \] where \( T \) is the number of T-shirts sold after \( n \) weeks.
- 20: Initial T-shirts sold in the first week.
- \( 1.2 \): Represents the increase based on the 20% (100% + 20%).
- \( n \): Represents the number of weeks passed.
Question 3
Linear Equation for Total Revenue: The linear equation representing total revenue \( y \) generated from selling \( x \) T-shirts at $20 each is: \[ y = 20x \]
Profit Calculation and Equation Modification: To find total profit, we need to account for costs. The total cost per T-shirt is $5 (bulk) + $5 (customization) + $2 (mailing) = $12. Adding the fixed cost of $50 for the workspace, the profit equation becomes: \[ y = 20x - (12x + 50) = 8x - 50 \]
Graph of Total Profit Equation:
- To graph the equation \( y = 8x - 50 \), plot the line where your x-intercept (where \( y = 0 \)) is at \( x = 6.25 \) (i.e., they must sell at least 7 T-shirts to make a profit).
Break-even Point: The friends must sell at least 7 T-shirts per week to start seeing profit because they would then earn $8 in profit per T-shirt.
Total Profit for Selling 25 T-shirts: Calculating for \( x = 25 \): \[ y = 8(25) - 50 = 200 - 50 = 150 \] Thus, the total profit is $150. Each friend’s share would then be: \[ \frac{150}{5} = 30 \] So, each friend would get $30.
Question 4
Exponential Growth Equation: The exponential growth of website visitors can be represented as: \[ y = 15(1.25^x) \] where \( y \) is the estimated number of visitors after \( x \) days.
Graph of Exponential Equation: To graph the equation:
- Plot points for each day:
- Day 0: 15 visitors (initial)
- Day 1: 15 * 1.25 = 18.75
- Continue for more days.
Visitors After 2 Weeks: The number of visitors expected after 2 weeks (14 days): \[ y = 15(1.25^{14}) \approx 15(8.26) \approx 123.9 \] Thus, about 124 visitors can be expected after 2 weeks.
Question 5
Growth Option Analysis:
- Option 1 - 5% Simple Interest leads to linear growth.
- Option 2 - 4.5% Compound Interest monthly leads to exponential growth.
- Option 3 - 7% Compound Interest annually also leads to exponential growth.
Interest Calculations After 5 Years:
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Option 1: \[ \text{Interest} = P \cdot r \cdot t = 275 \cdot 0.05 \cdot 5 = \text{$68.75} \] Final balance: \[ 275 + 68.75 = 343.75 \]
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Option 2 (using the formula for compound interest): \[ A = P(1 + \frac{r}{n})^{nt} = 275(1 + \frac{0.045}{12})^{12 \cdot 5} \approx 275(1.2330) \approx 339.08 \] Interest earned: \[ 339.08 - 275 = \text{64.08} \]
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Option 3: \[ A = P(1 + r)^t = 275(1.07)^5 \approx 275(1.40255) \approx 385.70 \] Interest earned: \[ 385.70 - 275 = 110.70 \]
Conclusion:
- Option 1 offers a final balance of $343.75.
- Option 2 offers a final balance of $339.08.
- Option 3 offers a final balance of $385.70.
Recommendation: The friends should choose Option 3 (7% compound interest annually) as it results in the highest return after 5 years, making their investment yield the most profit.
1 answer