Suppose that the sales at Borders bookstores went from 70 million dollars in 1989 to 415 million dollars in 1993. What is the continuous percent growth rate, per year, of sales?

let sales be

S = 70 e^kt, where S in in millions and t is the number of years since 1989

so when t=4 , S = 415

415 = 70 e^(4k)
5.92857 = e^4k
4k = ln 5.92857
k = ln 5.92857)/4 = .4449

or 44.49% continuous growth.

To find the continuous percent growth rate per year, we can use the formula:

r = ln(S2/S1)/(t2-t1)

where:
r is the continuous percent growth rate per year,
S1 is the initial value (70 million dollars in 1989),
S2 is the final value (415 million dollars in 1993),
t1 is the initial year (1989), and
t2 is the final year (1993).

Let's calculate the growth rate step-by-step:

1. Calculate the natural logarithm of the ratio of final value to initial value:
ln(415/70)

Using a calculator, we get:
ln(415/70) ≈ 1.996

2. Calculate the difference in years:
1993 - 1989 = 4

3. Divide the natural logarithm by the number of years to find the growth rate:
r = 1.996/4

Calculating the division, we get:
r ≈ 0.499

Therefore, the continuous percent growth rate, per year, of sales is approximately 0.499.

To find the continuous percent growth rate, we can use the formula:

Continuous percent growth rate = ln(final value / initial value) / (final year - initial year)

In this case:
- The initial value is $70 million in 1989.
- The final value is $415 million in 1993.
- The initial year is 1989.
- The final year is 1993.

Now, let's plug in these values into the formula and calculate step by step:

1. Calculate the value inside the natural logarithm:
ln(415,000,000 / 70,000,000)

2. Simplify the numerator:
ln(5.93)

3. Calculate the value of the natural logarithm using a calculator:
1.7755

4. Calculate the number of years between 1989 and 1993:
1993 - 1989 = 4

5. Divide the numerator by the number of years:
1.7755 / 4

6. Calculate the final annual continuous percent growth rate:
0.443875

Therefore, the continuous percent growth rate per year for Borders bookstores' sales during this period is approximately 0.443875 or 44.3875%.