The annual depreciation rate $r(0

Question

The annual depreciation rate $r(0<r<1)$ of a car purchased for $P$ dollars and worth A dollars after $t$ years can be modeled by the following formula:
$$\log (1-r)=\frac{1}{t} \log \frac{A}{P}$$
Find the depreciation rate of a car that is purchased for $\$ 35,000$ and is sold 3 years later for $\$ 22,000$. Express your answer as a percentage, and round the answer to the nearest whole percentage.
A) $14 \%$
B) $86 \%$
C) $-14 \%$
D) $-86 \%$

Quizplus · Accepted Answer

To find the depreciation rate $r$, we substitute $P = 35000$, $A = 22000$, and $t = 3$ into the given formula and solve for $r$:$$\log (1-r)=\frac{1}{3} \log \frac{22000}{35000}$$$$\log (1-r)=\frac{1}{3} \log \frac{22}{35}$$$$\log (1-r)=\frac{1}{3} \log \left(\frac{22}{35}\right)$$$$1-r = 10^{\left(\frac{1}{3} \log \left(\frac{22}{35}\right)\right)}$$$$r = 1 - 10^{\left(\frac{1}{3} \log \left(\frac{22}{35}\right)\right)}$$Using a calculator to evaluate the right-hand side gives:$$r \approx 0.14$$Expressed as a percentage and rounded to the nearest whole percentage, the depreciation rate is approximately $14\%$.

The Annual Depreciation Rate r(0<r<1)r(0<r<1)r(0<r<1) Of a Car Purchased For PPP Dollars and Worth a Dollars After

The Annual Depreciation Rate $r(0<r<1)$ Of a Car Purchased For $P$ Dollars and Worth a Dollars After