The Correct Answer and Explanation is:
The correct answer is f(x) = 42000(0.925)ˣ.
This problem involves modeling exponential decay, which describes a quantity decreasing at a rate proportional to its current value. The general formula for exponential growth or decay is f(x) = A(b)ˣ, where A is the initial amount, b is the growth or decay factor, and x is the number of time periods.
In this specific scenario, we are given:
- The initial value of the car, which is our starting amount, A. Here, A = $42,000.
- The rate of depreciation, which is 7.5% per year. Depreciation means the value is decreasing, so we are dealing with exponential decay.
To find the correct formula, we need to determine the decay factor, ‘b’. The decay factor is calculated as (1 – r), where ‘r’ is the rate of decay expressed as a decimal.
First, we must convert the percentage rate into a decimal:
r = 7.5% = 7.5 / 100 = 0.075
Next, we calculate the decay factor ‘b’:
b = 1 – r = 1 – 0.075 = 0.925
This decay factor of 0.925 signifies that the car retains 92.5% of its value each year, which is equivalent to losing 7.5% of its value.
Now, we can substitute the initial value (A = 42000) and the decay factor (b = 0.925) into the general exponential formula:
f(x) = A(b)ˣ
f(x) = 42000(0.925)ˣ
Let’s examine why the other options are incorrect:
- f(x) = 42000(1.075)ˣ: This formula models exponential growth, not decay, because the base (1.075) is greater than 1. This would represent the car’s value increasing by 7.5% each year.
- f(x) = 42000(.075)ˣ: This formula incorrectly uses the decimal rate (0.075) as the decay factor. This would imply a depreciation of 92.5% each year (1 – 0.075), which is a much faster loss of value than stated.
- f(x) = 75(1.42)ˣ: This formula uses an incorrect initial value ($75) and models exponential growth at a rate of 42%.
Therefore, the only formula that correctly models the car’s worth starting at $42,000 and depreciating by 7.5% annually is f(x) = 42000(0.925)ˣ.
