Respuesta :
The formula for compound growth is:
[tex]P=P_{0}(1+\frac{r}{n})^{nt}[/tex]
Where,
- P is the future value
- [tex]P_{0}[/tex] is the initial deposit
- r is the annual rate of interest (in decimal)
- n is the number of times compounding happens in a year (annual means n=1, compounding semi-annually means n=2, compounding quarterly means n=4 etc.)
- t is time in years
We need to solve for P. From the given information, we can write:
[tex]P_{0}=32,000[/tex]
[tex]r=\frac{3}{100}=0.03[/tex]
[tex]n=4[/tex] (since compounded quarterly)
[tex]t=7[/tex]
Plugging in all the information in the equation, we get:
[tex]P=32,000(1+\frac{0.03}{4})^{(4)(7)} \\P=32,000(1.0075)^{28}\\P=32,000(1.2327)\\P=39,446.4[/tex]
Alicia's account balance at end of 7 years is $39,446.4.
ANSWER: $39,446 (rounded to the nearest dollar)
The interest earned on the amount will increase the net balance. Thus, the balance of Alicia's account in 7 years would become $39,446.78 approximately.
How to calculate compound interest's amount?
If the initial amount (also called as principal amount) is P, and the interest rate is R% per unit time, and it is left for T unit of time for that simple interest, then the interest amount earned is given by:
[tex]CI = P(1 +\dfrac{R}{100})^T - P[/tex]
And thus, final amount will be:
[tex]A = P + CI = P(1 +\dfrac{R}{100})^T[/tex]
For the given case, the principal amount (initial amount) Lynne invested is P = 32000 (in dollars)
The rate of interest = 3% annual interest compounding quarterly.
or 3%/4 = 0.75% quarterly interest compounding quarterly. = R
Here, unit of time is is quarter of an year which means 1/4 of an year.
The time is given to be 7 years. In each year, there are 4 quarters, so in 7 years, there would be [tex]7 \times 4 = 28[/tex] quarters of years.
Thus, T = 28 (we always need to keep unit of time same for R and T for using that formula).
Putting these values in the above formula, we get the final amount's value in Alicia's account in 28 years as:
[tex]A = P(1 + \dfrac{R}{100})^T = 32000 ( 1 + \dfrac{0.75}{100})^{28} = 32000(1.0075)^{28}\\\\A \approx 39446.78 \: \rm \text{(in dollars)}[/tex]
Thus, the balance of Alicia's account in 7 years would become $39,446.78 approximately.
Learn more about compound interest here:
https://brainly.com/question/11897800
