Answer :
Certainly! Let's break down the problem step by step.
### Part A: Annual Rate of Change Between 1995 and 2005
To determine the annual rate of change, we use the formula for the geometric rate of change between two values over a certain period. The formula is:
[tex]\[ r = \left(\frac{{\text{Final Value}}}{{\text{Initial Value}}}\right)^{\frac{1}{n}} - 1 \][/tex]
Where:
- The Initial Value is \$32,000 (the car's value in 1995).
- The Final Value is \$13,000 (the car's value in 2005).
- \( n \) is the number of years between the two values, which is \( 2005 - 1995 = 10 \) years.
Using this formula with the given values:
[tex]\[ r = \left(\frac{13000}{32000}\right)^{\frac{1}{10}} - 1 \][/tex]
Computing this gives us the annual rate of change:
[tex]\[ r \approx -0.0861 \][/tex]
So, the annual rate of change, rounded to 4 decimal places, is:
[tex]\[ r = -0.0861 \][/tex]
### Part B: Annual Rate of Change in Percentage Form
To convert the decimal form of the annual rate of change to a percentage, we multiply by 100:
[tex]\[ r \times 100 \][/tex]
Doing this calculation:
[tex]\[ -0.0861 \times 100 = -8.6141\% \][/tex]
So, the annual rate of decrease in percentage form, rounded to four decimal places, is:
[tex]\[ r = -8.6141\% \][/tex]
### Part C: Projected Value in the Year 2008
Given that the annual depreciation rate continues to be the same, we can project the car's value for the year 2008. We will calculate the rate using 3 years (from 2005 to 2008) and the known annual rate of -0.0861.
The formula for future value is:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + r)^n \][/tex]
Where:
- Present Value = \$13,000 (the car's value in 2005).
- \( r = -0.0861 \) (the annual rate of change).
- \( n = 3 \) (the number of years from 2005 to 2008).
Using the formula and the given values:
[tex]\[ \text{Future Value} = 13000 \times (1 - 0.0861)^3 \][/tex]
Computing this gives us the value in 2008:
[tex]\[ \text{Future Value} \approx 9900\text{ (rounded to nearest fifty dollars)} \][/tex]
So, the car's projected value in 2008, rounded to the nearest 50 dollars, is:
[tex]\[ \text{Value} = \$9900 \][/tex]
To summarize:
- The annual rate of change is: \( r = -0.0861 \).
- The annual rate of change in percentage form is: \( r = -8.6141\% \).
- The projected value of the car in 2008 is: \$9900.
### Part A: Annual Rate of Change Between 1995 and 2005
To determine the annual rate of change, we use the formula for the geometric rate of change between two values over a certain period. The formula is:
[tex]\[ r = \left(\frac{{\text{Final Value}}}{{\text{Initial Value}}}\right)^{\frac{1}{n}} - 1 \][/tex]
Where:
- The Initial Value is \$32,000 (the car's value in 1995).
- The Final Value is \$13,000 (the car's value in 2005).
- \( n \) is the number of years between the two values, which is \( 2005 - 1995 = 10 \) years.
Using this formula with the given values:
[tex]\[ r = \left(\frac{13000}{32000}\right)^{\frac{1}{10}} - 1 \][/tex]
Computing this gives us the annual rate of change:
[tex]\[ r \approx -0.0861 \][/tex]
So, the annual rate of change, rounded to 4 decimal places, is:
[tex]\[ r = -0.0861 \][/tex]
### Part B: Annual Rate of Change in Percentage Form
To convert the decimal form of the annual rate of change to a percentage, we multiply by 100:
[tex]\[ r \times 100 \][/tex]
Doing this calculation:
[tex]\[ -0.0861 \times 100 = -8.6141\% \][/tex]
So, the annual rate of decrease in percentage form, rounded to four decimal places, is:
[tex]\[ r = -8.6141\% \][/tex]
### Part C: Projected Value in the Year 2008
Given that the annual depreciation rate continues to be the same, we can project the car's value for the year 2008. We will calculate the rate using 3 years (from 2005 to 2008) and the known annual rate of -0.0861.
The formula for future value is:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + r)^n \][/tex]
Where:
- Present Value = \$13,000 (the car's value in 2005).
- \( r = -0.0861 \) (the annual rate of change).
- \( n = 3 \) (the number of years from 2005 to 2008).
Using the formula and the given values:
[tex]\[ \text{Future Value} = 13000 \times (1 - 0.0861)^3 \][/tex]
Computing this gives us the value in 2008:
[tex]\[ \text{Future Value} \approx 9900\text{ (rounded to nearest fifty dollars)} \][/tex]
So, the car's projected value in 2008, rounded to the nearest 50 dollars, is:
[tex]\[ \text{Value} = \$9900 \][/tex]
To summarize:
- The annual rate of change is: \( r = -0.0861 \).
- The annual rate of change in percentage form is: \( r = -8.6141\% \).
- The projected value of the car in 2008 is: \$9900.
