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Any amount over 1,000 should include the comma (e.g., [tex]\$2,354.67[/tex]). All percentage values in the answers need to include the percentage (%) sign. For all items without specific rounding instructions, round your answers to two decimal places (e.g., 5.06).

A graphing calculator has determined this exponential regression equation based upon car value data: [tex]y = a \cdot b^x[/tex], where [tex]a = 18,547.23[/tex] and [tex]b = 0.8625[/tex].

1. What is the rate of depreciation for this car?
2. How much is this car worth after 6 years and 7 months?



Answer :

GinnyAnswer
Sure, let's solve this step-by-step, considering each part of the question:

1. Exponential Regression Equation:
Given the equation \( y = a \times b^x \) with \( a = 18,547.23 \) and \( b = 0.8625 \).

2. Calculation of Rate of Depreciation:
The formula for annual depreciation rate is \( (1 - b) \times 100 \).

- Here, \( b = 0.8625 \)
- \( 1 - 0.8625 = 0.1375 \)
- \( 0.1375 \times 100 = 13.75 \)

Hence, the rate of depreciation is \( 13.75\% \).

3. Value of the Car After 6 Years:
We use the exponential regression equation to find the car's value after 6 years.

- Substitute \( x = 6 \) into the equation \( y = 18547.23 \times 0.8625^6 \)

The car's value after 6 years is \$7,635.43.

4. Value of the Car After 6 Years and 7 Months:
First, convert 6 years and 7 months to total years.

- 7 months is \( \frac{7}{12} \) years.
- Add this to 6 years: \( 6 + \frac{7}{12} = 6.5833 \) years (rounded to four decimal places for accuracy).

Now, substitute \( x = 6.5833 \) into the equation \( y = 18547.23 \times 0.8625^{6.5833} \)

The car's value after 6 years and 7 months is \$7,004.22.

Let's summarize:

- Rate of Depreciation: \( 13.75\% \)
- Value After 6 Years: \$7,635.43
- Value After 6 Years and 7 Months: \$7,004.22

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