Answer:
Step-by-step explanation:
Notice that with the percent growth, each year the company is grows by 50% of the
current year’s total, so as the company grows larger, the number of stores added in a year
grows as well.
To try to simplify the calculations, notice that after 1 year the number of stores for
company B was:
100 + 0.50(100) or equivalently by factoring
100(1+ 0.50) = 150
We can think of this as “the new number of stores is the original 100% plus another
50%”.
After 2 years, the number of stores was:
150 + 0.50(150) or equivalently by factoring
150(1+ 0.50) now recall the 150 came from 100(1+0.50). Substituting that,
100(1 0.50)(1 0.50) 100(1 0.50) 225 2 + + = + =
After 3 years, the number of stores was:
225 + 0.50(225) or equivalently by factoring
225(1+ 0.50) now recall the 225 came from 2 100(1+ 0.50) . Substituting that,
100(1 0.50) (1 0.50) 100(1 0.50) 337.5 2 3 + + = + =
From this, we can generalize, noticing that to show a 50% increase, each year we
multiply by a factor of (1+0.50), so after n years, our equation would be n B(n) = 100(1+ 0.50)
In this equation, the 100 represented the initial quantity, and the 0.50 was the percent
growth rate. Generalizing further, we arrive at the general form of exponential functions.
Exponential Function
An exponential growth or decay function is a function that grows or shrinks at a
constant percent growth rate. The equation can be written in the form x f (x) = a(1+ r) or x f (x) = ab where b = 1+r
Where
a is the initial or starting value of the function
r is the percent growth or decay rate, written as a decimal
b is the growth factor or growth multiplier. Since powers of negative numbers behave
strangely, we limit b to positive values.
To see more clearly the difference between exponential and linear growth, compare the
two tables and graphs below, which illustrate the growth of company A and B described
above over a longer time frame if the growth patterns were to continueExample 2
A certificate of deposit (CD) is a type of savings account offered by banks, typically
offering a higher interest rate in return for a fixed length of time you will leave your
money invested. If a bank offers a 24 month CD with an annual interest rate of 1.2%
compounded monthly, how much will a $1000 investment grow to over those 24
months?
First, we must notice that the interest rate is an annual rate, but is compounded monthly,
meaning interest is calculated and added to the account monthly. To find the monthly
interest rate, we divide the annual rate of 1.2% by 12 since there are 12 months in a
