Creating an Exponential Model
In this activity, you will formulate and solve an exponential equation that models a real-world situation.
Emma doesn't have experience using credit cards. In fact, she just got her first one. She is also about to start her first year of college. She uses her new credit card to purchase textbooks for her classes. The total comes to $300. These are the terms of her credit card:
- It has a $15 \% annual interest rate.
- The interest is compounded monthly.
- The card has $0 minimum payments for the first four years it is active.
The expression that models this situation is [tex]P\left(1+\frac{r}{n}\right)^{n t}[/tex], where [tex]P[/tex] represents the initial, or principal, balance; [tex]r[/tex] represents the interest rate; [tex]t[/tex] represents the time in years; and [tex]n[/tex] represents the number of times the interest is compounded per year.
Part A
Question
Identify the values of [tex]P, r[/tex] and [tex]n[/tex] in the expression [tex]P\left(1+\frac{r}{n}\right)^{n t}[/tex] based on Emma's situation. Then substitute those values into the formula to write a simplified exponential expression in terms of time.
Replace the variables [tex]a, b[/tex], and [tex]c[/tex] to write the expression.
Answer (2)
Identify the principal balance P = $300 , the annual interest rate r = 0.15 , and the number of times interest is compounded per year n = 12 .
Substitute these values into the exponential growth formula: P ( 1 + n r ) n t = 300 ( 1 + 12 0.15 ) 12 t .
Simplify the expression: 300 ( 1.0125 ) 12 t .
The simplified exponential expression is 300 ( 1.0125 ) 12 t .
Explanation
Identifying the values of P, r, and n We are given the formula for exponential growth: P ( 1 + n r ) n t , where:
P is the principal balance,
r is the annual interest rate,
n is the number of times interest is compounded per year,
t is the time in years.
From the problem statement, we have:
Emma's initial balance, P = $300
The annual interest rate, r = 15% = 0.15
The interest is compounded monthly, so n = 12
Substituting the values into the formula and simplifying Now, we substitute these values into the formula: 300 ( 1 + 12 0.15 ) 12 t Next, we simplify the fraction inside the parentheses: 12 0.15 = 0.0125 So the expression becomes: 300 ( 1 + 0.0125 ) 12 t Which simplifies to: 300 ( 1.0125 ) 12 t
Final Simplified Expression Therefore, the simplified exponential expression in terms of time t is: 300 ( 1.0125 ) 12 t
Examples
Exponential models are used in various real-world situations, such as calculating the growth of a population, the decay of a radioactive substance, or the accumulation of interest in a savings account. In Emma's case, understanding how her credit card balance grows over time can help her make informed decisions about managing her debt and avoiding excessive interest charges. By using the exponential model, she can project her balance at any point in the future and plan her payments accordingly. This knowledge empowers her to take control of her finances and make responsible choices.
The values for Emma's credit card situation are: principal balance P = 300 , annual interest rate r = 0.15 , and monthly compounding n = 12 . Substituting these into the exponential model gives 300 ( 1.0125 ) 12 t as the final expression. This formula helps project her credit card balance as time goes on, considering the effect of compounded interest.
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