Let's analyze the depreciation functions provided:
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The car's depreciation function is given by: \[ y = 21,000(1 - 0.15)^x \] Here, the depreciation rate of the car is \( 0.15 \), which is 15%.
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The motorcycle's depreciation function is given by: \[ y = 14,000(1 - 0.09)^x \] Here, the depreciation rate of the motorcycle is \( 0.09 \), which is 9%.
Now, let's determine how many years it will take for each vehicle to depreciate to a value of $7,000.
For the Car:
We set the equation equal to $7,000 and solve for \( x \): \[ 7,000 = 21,000(1 - 0.15)^x \]
Divide both sides by 21,000: \[ \frac{7,000}{21,000} = (1 - 0.15)^x \] \[ \frac{1}{3} = (0.85)^x \]
Taking the logarithm of both sides: \[ \log\left(\frac{1}{3}\right) = x \log(0.85) \]
Now solve for \( x \): \[ x = \frac{\log\left(\frac{1}{3}\right)}{\log(0.85)} \]
Calculating: \[ \log(0.3333) \approx -0.4771 \] \[ \log(0.85) \approx -0.0706 \] \[ x \approx \frac{-0.4771}{-0.0706} \approx 6.75 \]
So, the car will be worth $7,000 after approximately 7 years.
For the Motorcycle:
We set the motorcycle's depreciation function equal to $7,000 and solve for \( x \): \[ 7,000 = 14,000(1 - 0.09)^x \]
Divide both sides by 14,000: \[ \frac{7,000}{14,000} = (1 - 0.09)^x \] \[ \frac{1}{2} = (0.91)^x \]
Taking the logarithm of both sides: \[ \log\left(\frac{1}{2}\right) = x \log(0.91) \]
Now solve for \( x \): \[ x = \frac{\log\left(\frac{1}{2}\right)}{\log(0.91)} \]
Calculating: \[ \log(0.5) \approx -0.3010 \] \[ \log(0.91) \approx -0.0430 \] \[ x \approx \frac{-0.3010}{-0.0430} \approx 6.99 \]
So, the motorcycle will be worth $7,000 after approximately 7 years.
Summary of Answers:
- The depreciation rate of the car is 15%.
- The depreciation rate of the motorcycle is 9%.
- The car will be worth $7,000 after approximately 7 years.
- The motorcycle will be worth $7,000 after approximately 7 years.