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.

### Answers

A = Ao x (1+r)^t

where

t = number of units of time.

A = the amount after "t" units of time.

Ao = The initial amount

Hence the final equation after substituting :

A = 3500 x (1+r)^10

(a)

(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

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

Initial value is 3500, it means a=3500.

f(x)=2000 at x=2.

The exponential function for the graph is

(b)

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

Substitute x=9.5 in the above function.

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

y = 3500 e^(-k⋅ 9.5)

Step-by-step explanation:

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.

Explanation:

Exponential equation is given by

y = 3500 ⋅ e^( k 9.5)

whereby:

y : value

A : constant;

k : rate of change

t : time value

In this when t =0

3500= A ⋅ e^ k 0

3500 = A

after 10 years we have

, after 9.5 years, the value of the boat is:

y = 3500 e^(-k⋅ 9.5)

k is the rate of change and it shows that it is negative because there is a depreciation in value. Note that the rate of change is not given in this case.

V(n) = a * b^n, where V(n) shows the value of boat after n years.

V(0) = 3500

V(2) = 2000

n = 0

V(0) = a * b^0 = 3500

a = 3500

V(2) = a * b^2

2000 = 3500 * b^2

b = sqrt (2000/3500)

b ≈ 0.76

V(n) = 3500 * 0.76^n

We can check it for n = 1 which is close to 2500 in the graph:

V(1) = 3500 * (0.76)^1

V(1) = 2660

And in the graph we have V(3) ≈ 1500,

V(n) = 3500 * (0.76)^3 ≈ 1536

Now n = 9.5

V(9.5) = 3500 * (0.76)^(9.5)

V(9.5) ≈ 258