Math Problem Statement
Express the amount A in the account as a function of the term of the investment in t years. a. A=2000(1+.7125/12)^12t b. A=2000(1+7.125/12)^12t c. A=2000(1+.07125/12)^12t d. A=2000(1+.07125)^tHow much is in the account after 5 years? Type a response Required 3 points 3 Question 13 13. How long will it take for the initial investment to double? Round to 2 decimal places
Solution
Let's break down the problem step by step.
Given Information:
The equation for the amount in the account is related to the compound interest formula, where:
- is the initial principal,
- is the annual interest rate,
- is the number of compounding periods per year,
- is the time in years.
The general form of the compound interest formula is:
Option a:
- Initial Principal () = 2000
- Annual Interest Rate () = 0.07125 (or 7.125%)
- Number of Compounding Periods per Year () = 12 (monthly compounding)
This equation represents compound interest with monthly compounding. The formula is:
a) Amount in the account after 5 years:
To find the amount after 5 years, substitute into the equation:
Let’s calculate it:
So, after 5 years, the amount in the account will be approximately $2851.52.
b) Time required for the investment to double:
To find how long it takes for the initial investment to double, we set , meaning the amount in the account is double the initial investment. So, the equation becomes:
Simplify:
Now, take the natural logarithm (ln) of both sides:
Solve for :
So, it will take approximately 9.73 years for the initial investment to double.
Recap of the Answers:
- After 5 years, the amount in the account is approximately $2851.52.
- Time to double the investment is approximately 9.73 years.
Do you need any further clarification or additional details?
Additional Questions:
- What is the impact of increasing the interest rate on the time to double the investment?
- How would the amount in the account change if compounding were done quarterly instead of monthly?
- If the initial investment were $2500, how would this affect the final amount after 5 years?
- How can the formula be adjusted if the interest is compounded continuously instead of periodically?
- How would the time to double the investment change if the annual interest rate were 10% instead of 7.125%?
Tip:
The more frequently the interest is compounded, the more interest accumulates over time due to the effects of compounding.
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Exponential Growth
Logarithms
Formulas
Compound Interest Formula: A = P(1 + r/n)^(nt)
Doubling Time Formula: 2 = (1 + r/n)^(nt)
Theorems
Logarithmic Properties
Exponential Growth Theorem
Suitable Grade Level
Grades 10-12
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