Business: The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10 years a. write an exponential function for the graph. b. use the function in part a to find the value of the boat after 9.5 years.

The exponential decay graph shows the expected depreciation

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09.01.2022

(a) $f(x)=3500(\frac{2}{\sqrt{7}})^x$

(b) $245.27

Explanation:

(a)

From the below graph it is clear that the graph it passes through the points (0,3500) and (2,2000).

The general form of an exponential function is

f(x)=ab^x

where, a is the initial value and b is growth or decay factor.

Initial value is 3500, it means a=3500.

f(x)=3500b^x

f(x)=2000 at x=2.

2000=3500b^2

$\frac{2000}{3500}=b^2$

$\frac{4}{7}=b^2$

$\sqrt{\frac{4}{7}}=b$

$\frac{2}{\sqrt{7}}=b$

The exponential function for the graph is

$f(x)=3500(\frac{2}{\sqrt{7}})^x$

(b)

We need to find the value of the boat after 9.5 years.

Substitute x=9.5 in the above function.

$f(9.5)=3500(\frac{2}{\sqrt{7}})^{9.5}$

f(9.5)=245.26598

$f(9.5)\approx 245.27$

Therefore, the value of the boat after 9.5 years is $245.27.

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