3400 dollars is placed in an account with an annual interest rate of 7-5%. How much
will be in the account after 17 years, to the nearest cent?
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attematizantofel
The Correct answer and Explanation is:
To calculate how much will be in the account after 17 years, we need to use the formula for compound interest. The compound interest formula is:A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}A=P(1+nr)nt
Where:
- AAA is the amount of money in the account after interest.
- PPP is the principal amount (initial investment), which in this case is $3400.
- rrr is the annual interest rate in decimal form. Since the annual interest rate is 7.5%, r=0.075r = 0.075r=0.075.
- nnn is the number of times the interest is compounded per year. For this calculation, we’ll assume the interest is compounded annually, meaning n=1n = 1n=1.
- ttt is the number of years the money is invested, which is 17 years.
Step-by-step calculation:
- Principal (P): $3400
- Interest rate (r): 7.5% or 0.075 in decimal.
- Number of times interest compounded per year (n): 1 (annually).
- Time (t): 17 years.
Now, substitute the values into the compound interest formula:A=3400(1+0.0751)1×17A = 3400 \left(1 + \frac{0.075}{1}\right)^{1 \times 17}A=3400(1+10.075)1×17
This simplifies to:A=3400(1+0.075)17A = 3400 \left(1 + 0.075\right)^{17}A=3400(1+0.075)17 A=3400(1.075)17A = 3400 \left(1.075\right)^{17}A=3400(1.075)17
Now, calculate 1.075171.075^{17}1.07517:1.07517≈3.3791.075^{17} \approx 3.3791.07517≈3.379
Next, multiply by the principal:A=3400×3.379≈11488.6A = 3400 \times 3.379 \approx 11488.6A=3400×3.379≈11488.6
Thus, the amount in the account after 17 years will be $11,488.60.
Explanation:
Compound interest grows money faster than simple interest because it calculates interest on both the initial principal and the accumulated interest. This leads to exponential growth over time. In this example, the $3400 is growing at an annual rate of 7.5%, and after 17 years, the total amount becomes $11,488.60. This illustrates the power of compound interest and how investments can grow significantly over long periods, even with a modest interest rate.