Exponential decay

[Guest]

The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10 years.

A coordinate graph is shown. The horizontal axis extends from 0 to 12 years. The vertical axis extends from 0 to 9500 with an axis label of 'Value' in dollars. A curve is graphed which begins at 0 comma 3500, then decreases passing through approximately 1 comma 2700

Write an exponential function for the graph. Use the function to find the value of the boat after 9.5 years.

[HallsofIvy]

"Exponential decay" is, by definition, a function of the form [tex]y= Ca^x[/tex] for some numbers C and a (a< 1 for "decay"). To determine the function you need to determine the two values, a and C. To determine two numbers you need two equations.

You are told that the graph goes through (0, 3500) and (1, 2700). That is, when x= 0,l y= 3500 so [tex]y= Ca^x[/tex] becomes [tex]3500= Ca^0= C[/tex]. And when x= 1, y= 2700 so [tex]2700= Ca^1= Ca[/tex]. Knowing that C= 3500, [tex]2700= 3500a[/tex] so [tex]a= \frac{2700}{3500}= \frac{27}{35}[/tex].