a car depreciates an average of 7.5% annually. the purchase price of the car was $28,946. what equation to…

Explanation:

Step1: Recall the depreciation formula

The formula for exponential - depreciation is $A = P(1 - r)^t$, where $A$ is the final value, $P$ is the initial value, $r$ is the rate of depreciation, and $t$ is the number of years. Here, $P=$28946$, $r = 0.075$ (since $7.5%=0.075$), and $A = $25775.05$.

Step2: Substitute the values into the formula

We get $25775.05=28946(1 - 0.075)^t$, which simplifies to $25775.05=28946\times0.925^t$.

Step3: Solve for $t$

First, divide both sides of the equation by $28946$: $\frac{25775.05}{28946}=0.925^t$. So, $0.89045 = 0.925^t$. Take the natural logarithm of both sides: $\ln(0.89045)=t\ln(0.925)$. Then, $t=\frac{\ln(0.89045)}{\ln(0.925)}$. We know that $\ln(0.89045)\approx - 0.116$ and $\ln(0.925)\approx-0.078$. So, $t=\frac{- 0.116}{-0.078}\approx1.5$.

Answer:

$t\approx1.5$ years