Below is a situation that can be described using an exponential function. An investment of $1800 earns 6% annual interest, compounded monthly. What is an appropriate unit of time (... Below is a situation that can be described using an exponential function. An investment of $1800 earns 6% annual interest, compounded monthly. What is an appropriate unit of time (such as days, weeks, years)? What is the multiplier that should be used? What is the initial value? What is an exponential equation in the form y = a(b)^t that can be used to model the situation?
Understand the Problem
The question outlines a financial scenario involving an investment that earns compound interest. It seeks specific inputs related to time measurement, multipliers for growth, initial investment value, and the formulation of an exponential equation that models the investment growth.
Answer
- Appropriate unit of time: months - Multiplier used: 1.005 - Initial value: 1800 - Exponential equation: \( y = 1800(1.005)^t \)
Answer for screen readers
- Appropriate unit of time: months
- Multiplier used: 1.005
- Initial value: 1800
- Exponential equation: ( y = 1800(1.005)^t )
Steps to Solve
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Determine the Appropriate Unit of Time Since the interest is compounded monthly, the appropriate unit of time is months.
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Calculate the Multiplier for Growth The formula to determine the multiplier for compound interest is given by: [ b = 1 + \frac{r}{n} ] Where:
- $r$ is the annual interest rate (as a decimal)
- $n$ is the number of times interest is compounded per year
For this problem:
- $r = 0.06$
- $n = 12$
Substituting these values, we get: [ b = 1 + \frac{0.06}{12} = 1 + 0.005 = 1.005 ]
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Identify the Initial Value The initial investment value is given as: [ \text{Initial Value} = 1800 ]
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Formulate the Exponential Equation The general form of an exponential equation for compound interest is: [ y = a(b)^t ] Where:
- $a$ is the initial amount
- $b$ is the growth multiplier
- $t$ is the time in months
Substituting our known values: [ y = 1800(1.005)^t ]
- Appropriate unit of time: months
- Multiplier used: 1.005
- Initial value: 1800
- Exponential equation: ( y = 1800(1.005)^t )
More Information
This scenario demonstrates how compound interest accumulates over time. The multiplier of 1.005 means every month, the investment grows by 0.5%. The formula derived allows for calculating the total investment at any given month.
Tips
- Confusing the time unit: Ensure to use months since the interest is compounded monthly.
- Incorrectly calculating the multiplier: Remember to divide the annual rate by the number of compounding periods per year.
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