Question

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.

Answer 1

A. y=3500(2/√7)^x

B. y=$245.27

Answer 2

(a)
`f(x)=3500((2)/(√(7)))^x`

(b) $245.27

Step-by-step 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`

`(2000)/(3500)=b^2`

`(4)/(7)=b^2`

`\sqrt{(4)/(7)}=b`

`(2)/(√(7))=b`

The exponential function for the graph is

`f(x)=3500((2)/(√(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((2)/(√(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.