To determine the correct answers for the drop-down menus, let’s analyze the information provided.
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Initial Resale Value: The initial resale value for each family’s laptop is found by evaluating the functions at \(x = 0\).
For the Sawyer family’s laptop: \[ f(0) = 620(0.85)^0 = 620 \]
For the Ruiz family’s laptop: The resale values for years 0, 2, 4, and 6 are given. At year 0: \[ g(0) = 640 \]
Since \(g(0) = 640\) and \(f(0) = 620\), the Ruiz family’s laptop has the greater initial resale value.
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Average Rate of Decrease: To find the average rate of decrease in value for both families over the first four years, we will calculate the depreciation for both laptops.
Sawyer Family’s Laptop:
- Initial value at year 0: \(620\)
- Value at year 4: \(f(4) = 620(0.85)^4\)
- Calculate \(f(4)\): \[ f(4) = 620 \times (0.85)^4 \approx 620 \times 0.5220 \approx 324.84 \]
- Average rate of decrease over 4 years: \[ \text{Average rate} = \frac{620 - 324.84}{4} \approx \frac{295.16}{4} \approx 73.79 \]
Ruiz Family’s Laptop:
- Initial value at year 0: \(640\)
- Value at year 4: \(g(4) = 360\)
- Average rate of decrease over 4 years: \[ \text{Average rate} = \frac{640 - 360}{4} = \frac{280}{4} = 70 \]
Now we can compare the average rates:
- Sawyer family: approximately \(73.79\)
- Ruiz family: \(70\)
Since the average rate of decrease for the Sawyer family’s laptop is greater than that of the Ruiz family’s laptop, we can fill in the gaps accordingly.
Final answers:
- The Ruiz family’s laptop has the greater initial resale value.
- During the first four years, the resale value of the Sawyer family’s laptop decreases at an average rate greater than the resale value of the Ruiz family’s laptop.