A car has a purchase price of $24,800. The value declines continuously at an exponential...

Step 1: Consider the provided information, A car has purchase on price $24,800 and the value of car declines continuously at an exponential rate of 23% annually.

Step 2: (A) Consider the function for exponential decay is, $\Uparrow$
${\displaystyle P=P_0{(1-r\mathrm{\%})}^t}$
${\displaystyle =24800{(1-\frac{23}{100})}^t}$
${\displaystyle =24800{\left(\frac{77}{100}\right)}^t}$
${\displaystyle =24800{\left(0.77\right)}^t}$ Therefore, the exponential funktion is $P=24800(0.77{)}^t$.

Step 3: (B) Substitute t=6 in above function,
${\displaystyle P=24800{\left(0.77\right)}^6}$
${\displaystyle \approx \mathrm{\$}5168.88}$ Therefore, the prise after 6 years is $5168.88 Step 4: (c) Substitute P=2000 in the above function.
${\displaystyle 2000=24800{\left(0.77\right)}^t}$
${\displaystyle {\left(0.77\right)}^t=\frac{2000}{24800}}$
${\displaystyle {\left(0.77\right)}^t=\frac{5}{62}}$
$t\ln (0.77)=\ln (\frac{5}{65})$
${\displaystyle t=\frac{In\left(\frac{5}{62}\right)}{In\left(\frac{0}{77}\right)}}$
$\approx 9.6$ years Therefore, the time requarite for value of car become $2000 is 9.6 years.