writing an equation that models exponential growth or decay

Question

In 2010

, a car was purchased for $20,000

. Each year since, the resale value has decreased by 25%

.Let t

 be the number of years since 2010

. Let y

 be the value of the car, in dollars.Write an exponential function showing the relationship between y

 and t

.

Answer 1

For exponential growth we have the basic function y = A*b^nt, where A is the principle, b is the log base, and n is the rate of change over one unit of time (Units are t^-1, e.g. Hertz). We're going to choose n =1 here so we don't have to deal with it.

This question is really about finding where to plug in the values. The principle is the amount of money we start with, which here is $20,000. For the log base, we want to choose a value that decreases the principle by 25% of its previous value every year. So, in the same way we do with compound interest, we will pick

b = (1 +/- rate of gain or loss as a decimal)

For this example, if we were losing 25% of the value every year, we would choose the rate of change to be -0.25,

b = (1 - 0.25)

b = 0.75

So, we have arrived at our answer,

y = 20000 (0.75)^t

Let me know if that makes sense of you'd like another example!

-Ethan

Answer 2

When something changes by a percentage and then that result again changes by that percentage, etc, etc, that is an exponential situation. An equation for it is:

(Future Value) = (Initial Value)(1 +/- (rate in decimal form))^time or:

A(t) = A₀(1 +/- r)^t where A₀ is the value at time t = 0 and r is the rate in decimal form and t is the time that goes by. It's important that the units of the rate (r) match the units of the time (t), like "rate per year" and "years".

Please note that the equation has a +/- in it. You have to choose which one to use (you don't use both at the same time). If it's something like population that is growing, you use the "+". If it's population that's shrinking, you use the "-". In this case the value of the car is going down, so use the "-". So the equation is: A(t) = 20000(1 - 0.25)^t where t is in years. You can simplify what's in the parenthesis to 0.75 and they want "y" instead of "A(t)", so y = 20000(0.75)^t where t is years since 2010.