The equation A(t) = 22, 000(0. 85)' represents the value of a car t years after it was purchased.

Question

Which three statements are true of this situation?
• When new, the car cost $18,700.
• When new, the car cost $22,000
• The car's value is decreasing at a rate of 0.85% per year
• The car's value is decreasing at a rate of 85% per year
• The car's value is decreasing at a rate of 15% per year
• The car's value is decreasing at a rate of 0.15% per year
• In 5 years, the car's value will be about $9,762
• In 5 years, the car's value will be about $13,510

Answer 1

To analyze the equation \( A(t) = 22,000(0.85)^t \), we can deduce the following truths about the car's depreciation and its initial value:

1. When new, the car cost $22,000: This is true because \( A(0) = 22,000(0.85)^0 = 22,000 \).

2. The car's value is decreasing at a rate of 15% per year: This is true because the factor \( 0.85 \) indicates that the car retains 85% of its value each year, which means it loses 15% of its value each year.

Now let's evaluate the depreciation values and another statement:

3. In 5 years, the car's value will be about $13,510:

   • To find \( A(5) \): \[ A(5) = 22,000(0.85)^5 \] \[ A(5) \approx 22,000 \times 0.4420 \approx 9,724 \]

   Therefore, this statement about \( A(5) \) being about $13,510 is false since the value is approximately $9,724 (close to $9,762).

4. In 5 years, the car's value will be about $9,762: This is true based on the calculation above.

Evaluating all the statements, the three true statements are:

• When new, the car cost $22,000.
• The car's value is decreasing at a rate of 15% per year.
• In 5 years, the car's value will be about $9,762.