f(x)=18,000(.88)^x ; x represents the number of years since the car started to depreciate and 18,000 = the initial value of the car. What year will you recieve the car when the car value drops below $2,000? Please do a step by step
The Correct Answer and Explanation is:
We are given the function for the depreciation of the car’s value over time:
[
f(x) = 18,000 \cdot (0.88)^x
]
Where:
- ( f(x) ) represents the value of the car after ( x ) years,
- ( 18,000 ) is the initial value of the car,
- ( 0.88 ) is the depreciation factor (the car retains 88% of its value each year),
- ( x ) represents the number of years since the car started depreciating.
We need to find the value of ( x ) (the number of years) when the value of the car drops below $2,000.
Step 1: Set up the equation
We are looking for when ( f(x) < 2,000 ). This gives us the inequality:
[
18,000 \cdot (0.88)^x < 2,000
]
Step 2: Isolate the exponential expression
First, divide both sides of the inequality by 18,000 to isolate the exponential term:
[
(0.88)^x < \frac{2,000}{18,000}
]
Simplify the right side:
[
(0.88)^x < \frac{1}{9}
]
Step 3: Apply logarithms
To solve for ( x ), take the natural logarithm (ln) of both sides:
[
\ln((0.88)^x) < \ln\left(\frac{1}{9}\right)
]
Using the logarithmic rule ( \ln(a^b) = b \cdot \ln(a) ), we get:
[
x \cdot \ln(0.88) < \ln\left(\frac{1}{9}\right)
]
Step 4: Solve for ( x )
Now, calculate the natural logarithms:
- ( \ln(0.88) \approx -0.1278 )
- ( \ln\left(\frac{1}{9}\right) \approx -2.1972 )
Substitute these values into the equation:
[
x \cdot (-0.1278) < -2.1972
]
Now, divide both sides by -0.1278 (since dividing by a negative number reverses the inequality):
[
x > \frac{-2.1972}{-0.1278}
]
Calculate the right side:
[
x > 17.2
]
Step 5: Round to the nearest whole year
Since ( x ) represents the number of years, we round up to the next whole year, as the value drops below $2,000 after the 18th year.
Thus, the car’s value will drop below $2,000 in year 18.
Explanation:
This problem involves exponential depreciation, where the value of the car decreases by a fixed percentage each year. The model ( f(x) = 18,000 \cdot (0.88)^x ) describes how the value of the car decreases over time. The equation ( f(x) < 2,000 ) represents the point where the value drops below $2,000. By solving this inequality using logarithms, we can find that the car’s value will fall below $2,000 after 17.2 years. Since time is measured in whole years, we round up to the next full year, giving us an answer of 18 years. This demonstrates how exponential decay works in practical scenarios like depreciation.