Find the value of $800 deposited in a bank at 10% interest for 8 years compounded in the following ways. (Round your answers to the nearest cent.)
(a) annually_________$
(b) quarterly_________$
(c) continuously________$
To find the value of $800 deposited in a bank at 10% interest for 8 years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
(a) Annually:
In this case, the interest is compounded once per year, so n = 1.
A = 800(1 + 0.10/1)^(1*8)
Calculating,
A = 800(1 + 0.10)^8
A = 800(1.10)^8
A ≈ $1,954.24
Therefore, the value of $800 deposited in a bank at 10% interest and compounded annually for 8 years is approximately $1,954.24.
(b) Quarterly:
In this case, the interest is compounded four times per year, so n = 4.
A = 800(1 + 0.10/4)^(4*8)
Calculating,
A = 800(1 + 0.10/4)^32
A = 800(1.025)^32
A ≈ $1,975.62
Therefore, the value of $800 deposited in a bank at 10% interest and compounded quarterly for 8 years is approximately $1,975.62.
(c) Continuously:
In this case, the interest is compounded continuously.
A = P * e^(rt)
Where e is the base of the natural logarithm, which is approximately 2.71828.
A = 800 * e^(0.10*8)
Calculating,
A = 800 * e^(0.80)
A ≈ $2,013.75
Therefore, the value of $800 deposited in a bank at 10% interest and compounded continuously for 8 years is approximately $2,013.75.
To find the value of $800 deposited in a bank at 10% interest for 8 years compounded annually, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
(a) To calculate the value of $800 compounded annually, we plug in the given values:
P = $800
r = 10% (or 0.10 as a decimal)
n = 1 (compounded once per year)
t = 8 years
\[ A = 800 \left(1 + \frac{0.10}{1}\right)^{1 \cdot 8} \]
Simplifying this equation, we get:
\[ A = 800(1.10)^8 \]
Evaluating this expression, we find:
\[ A \approx \$1,855.07 \]
Therefore, the value of $800 compounded annually after 8 years is approximately $1,855.07.
(b) To find the value of $800 compounded quarterly, we need to adjust the formula.
Since interest is compounded quarterly, we divide the annual interest rate by 4 and multiply the number of years by 4.
P = $800
r = 10% (or 0.10 as a decimal)
n = 4 (compounded quarterly)
t = 8 years
\[ A = 800 \left(1 + \frac{0.10}{4}\right)^{4 \cdot 8} \]
Simplifying this expression, we get:
\[ A = 800(1.025)^{32} \]
Evaluating this equation, we find:
\[ A \approx \$1,862.95 \]
Therefore, the value of $800 compounded quarterly after 8 years is approximately $1,862.95.
(c) To find the value of $800 compounded continuously, we use the formula:
\[ A = Pe^{rt} \]
where:
P = $800
r = 10% (or 0.10 as a decimal)
t = 8 years
\[ A = 800e^{0.10 \cdot 8} \]
Evaluating this expression, we get:
\[ A \approx \$2,225.54 \]
Therefore, the value of $800 compounded continuously after 8 years is approximately $2,225.54.
800(1 + .10/n)^(8n) -- periodic compounding
or
800e^(.10*8) -- continuous compounding
I'm sure you already had the formulas, so just plug in your values for n as needed