devon purchased a new car valued at $16,000 t...
devon purchased a new car valued at $16,000 that depreciated continuously at a rate of 35%. its current value is $2,000. the equation 2,000 = 16,000(1 - r)^t represents the situation, where t is the age of the car in years and r is the rate of depreciation. about how old is devons car? use a calculator and round your answer to the nearest whole number. o 1 year o 2 years o 5 years o 8 years
Answer
# Explanation:
## Step1: Identify values and equation
The equation for depreciation is $V = P(1 - r)^t$, where $V = 2000$ (current value), $P=16000$ (initial value), and $r = 0.3$. So the equation becomes $2000=16000(1 - 0.3)^t$.
## Step2: Simplify the equation
First, divide both sides of the equation by 16000: $\frac{2000}{16000}=(0.7)^t$. Which simplifies to $0.125=(0.7)^t$.
## Step3: Take the natural - logarithm of both sides
$\ln(0.125)=t\ln(0.7)$.
## Step4: Solve for t
$t=\frac{\ln(0.125)}{\ln(0.7)}$. Using a calculator, $\ln(0.125)\approx - 2.079442$ and $\ln(0.7)\approx - 0.356675$. Then $t=\frac{-2.079442}{-0.356675}\approx5.83$. Rounding to the nearest whole number, $t = 6$. But since 6 is not an option and 5 is the closest among the given options.
# Answer:
C. 5 years