Question

Find the principal P that will generate the given future value​ A, where A=​$ 12000 at ​8% compounded daily for 10 years.

Answer 1

To find the principal amount, P, that will generate a future value, A, we can use the formula for compound interest: A = P(1 + r/n)^(nt) Where: A = Future value P = Principal amount r = Annual interest rate (in decimal form) n = Number of times the interest is compounded per year t = Number of years In this case, we are given: A = $12000 r = 8% = 0.08 n = 365 (compounded daily) t = 10 years Substituting these values into the formula, we have: 12000 = P(1 + 0.08/365)^(365*10) To solve for P, we need to isolate it on one side of the equation. Let's go through the steps: 1. Divide both sides of the equation by (1 + 0.08/365)^(365*10): 12000 / (1 + 0.08/365)^(365*10) = P 2. Calculate the right side of the equation: P ≈ $5,815.95 Therefore, the principal amount, P, that will generate a future value of $12000 at an interest rate of 8% compounded daily for 10 years is approximately $5,815.95.

Explanation:

Answer 2

$5,392.42

Explanation:

You want the principal P that will result in a future value A of $12,000 at interest rate 8% when compounded daily for 10 years.

Compound interest

The value of the account earning compound interest is ...

A = P(1 +r/n)^(nt)

where P is the principal invested at rate r, compounded n times per year for 10 years.

We want P when A = 12000, r = 0.08, t = 10, n = 365

P = 12000/(1 +0.08/365)^(365·10) ≈ 12000/2.2253458 ≈ 5392.42

The required principal is $5,392.42.

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Additional comment

The presence of 2 or 3 leap days in the 10 year period may change the required amount, depending on how they affect the daily rate.

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