Suppose that the sales at Borders bookstores went from 68 million dollars in 1990 to 427 million dollars in 1995 . Find an exponential function to model the sales (in millions of dollars) as a function of years,t, since 1990.

To find an exponential function to model the sales as a function of years, t, since 1990, we can use the general form of an exponential function:

y = a * b^t

where:
y is the sales in millions of dollars
t is the number of years since 1990
a is the initial sales value at t=0 (1990)
b is the growth rate factor

We are given that the sales in 1990 were 68 million dollars, so we can substitute y=68 and t=0 into the equation:

68 = a * b^0

Simplifying the equation, we get:

68 = a * 1

This implies that 'a' is 68.

Therefore, our equation becomes:

y = 68 * b^t

Now, we need to find the value of 'b' to complete the equation. We can use the second data point given, which states that sales in 1995 were 427 million dollars. In this case, t=5 since 1995 is 5 years after 1990. Substituting these values into the equation, we get:

427 = 68 * b^5

Dividing both sides of the equation by 68, we have:

6.279 = b^5

To find the value of 'b', we need to raise both sides of the equation to the power of 1/5:

(6.279)^(1/5) = (b^5)^(1/5)

Simplifying, we get:

1.444 ≈ b

Therefore, our final exponential function to model the sales as a function of years, t, since 1990 is:

y = 68 * (1.444)^t

To find an exponential function to model the sales as a function of years, we need to use the formula for exponential growth:

y = a * (1 + r)^t

Where:
- y represents the sales in millions of dollars
- a represents the initial amount (sales in 1990)
- r represents the growth rate per year
- t represents the time in years since 1990

We are given two data points:
- In 1990, the sales were 68 million dollars.
- In 1995, the sales were 427 million dollars.

Let's use the 1990 data point to find the initial amount, a.

68 = a * (1 + r)^0
68 = a

So, a = 68 million dollars.

Now, let's use the 1995 data point to find the growth rate, r.

427 = 68 * (1 + r)^5

To solve for r, divide both sides of the equation by 68:

(1 + r)^5 = 427/68
(1 + r)^5 = 6.2794

Take the fifth root of both sides of the equation:

1 + r ≈ (6.2794)^(1/5)
1 + r ≈ 1.3074

Subtract 1 from both sides of the equation:

r ≈ 0.3074

Now that we have the values for a and r, we can write the exponential function to model the sales:

y = 68 * (1 + 0.3074)^t

Simplifying this equation further:

y = 68 * (1.3074)^t

Therefore, the exponential function to model the sales as a function of years since 1990 is y = 68 * (1.3074)^t.

t = 0 in 1990

t = 5 in 1995

s = A e^kt
when t = 0, s = 68
so
68 = A

when t = 5
427 = 68 e^5k
6.28 = e^5k
ln 6.28 = 1.84 = 5 k
so
k = .367
so
s = 68 e^.367 t