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
