You invest $1000 at 7% interest compounded annually. What is the exponential equation that models this situation? According to your equation, how much will the investment be worth... You invest $1000 at 7% interest compounded annually. What is the exponential equation that models this situation? According to your equation, how much will the investment be worth 3 years later?
Understand the Problem
The question describes an investment of $1000 at a 7% annual interest rate, compounded annually. It asks for the exponential equation that models this scenario and then asks to calculate the value of the investment after 3 years based on that equation. We need to first write the equation and then substitute the variables to calculate the amount after 3 years.
Answer
$A = 1000(1.07)^t$ \$1225.04
Answer for screen readers
The exponential equation that models the investment is $A = 1000(1.07)^t$. After 3 years, the investment will be worth $1225.04.
Steps to Solve
- Write down the general formula for compound interest
The formula for annual compound interest is: $A = P(1 + r)^t$ Where: $A$ = the future value of the investment/loan, including interest $P$ = the principal investment amount (the initial deposit or loan amount) $r$ = the annual interest rate (as a decimal) $t$ = the number of years the money is invested or borrowed for
- Identify given values
From the problem: $P = 1000$ $r = 7% = 0.07$
- Plug in the values into the formula
Substitute $P$ and $r$ into the formula for compound interest: $A = 1000(1 + 0.07)^t$ $A = 1000(1.07)^t$
- Calculate the investment value after 3 years
Substitute $t = 3$ into the equation: $A = 1000(1.07)^3$
- Simplify the expression
Calculate $(1.07)^3$: $(1.07)^3 = 1.225043$ Multiply by 1000: $A = 1000 \times 1.225043 = 1225.043$
- Round to nearest cent Rounded to the nearest cent, the investment will be worth $1225.04
The exponential equation that models the investment is $A = 1000(1.07)^t$. After 3 years, the investment will be worth $1225.04.
More Information
The formula used in this problem is a fundamental concept in finance for calculating the future value of an investment with compound interest. Understanding how to apply this formula is useful for making informed financial decisions.
Tips
A common mistake is forgetting to convert the interest rate from a percentage to a decimal. For example, using 7 instead of 0.07 in the formula will lead to a dramatically incorrect result. Another mistake is incorrectly calculating the exponent or misunderstanding the order of operations.
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