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Math  /  Calculus

QuestionLamonte is going to invest in an account paying an interest rate of 4\% compounded continuously. How much would Lamonte need to invest, to the nearest cent, for the value of the account to reach \$12,300 in 8 years?

Studdy Solution

STEP 1

1. The interest is compounded continuously.
2. The future value of the investment is \$12,300.
3. The annual interest rate is 4\%.
4. The investment period is 8 years.

STEP 2

1. Recall the formula for continuous compounding.
2. Rearrange the formula to solve for the initial investment.
3. Substitute the given values.
4. Calculate the initial investment.

STEP 3

Recall the formula for continuous compounding:
A=Pert A = Pe^{rt}
where: - A A is the future value of the investment, - P P is the principal amount (initial investment), - r r is the annual interest rate (as a decimal), - t t is the time in years, - e e is the base of the natural logarithm.

STEP 4

Rearrange the formula to solve for the initial investment P P :
P=Aert P = \frac{A}{e^{rt}}

STEP 5

Substitute the given values into the formula:
A=12300 A = 12300 r=0.04 r = 0.04 t=8 t = 8
P=12300e0.04×8 P = \frac{12300}{e^{0.04 \times 8}}

STEP 6

Calculate the initial investment:
P=12300e0.32 P = \frac{12300}{e^{0.32}}
First, calculate e0.32 e^{0.32} :
e0.32≈1.377127764 e^{0.32} \approx 1.377127764
Then, calculate P P :
P=123001.377127764≈8929.68 P = \frac{12300}{1.377127764} \approx 8929.68
The initial investment Lamonte needs to make is:
8929.68 \boxed{8929.68}

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