Math  /  Algebra

Question16. Use the compound interest formulas to solve the following. Round answers to the nearest cent. a. Find the accumulated value of an investment of $25,000\$ 25,000 for 4 years at an interest rate of 7%7 \% if the money is compounded monthly. b. Find the accumulated value of an investment of $25,000\$ 25,000 for 4 years at an interest rate of 7%7 \% if the money is compounded countinuously.

Studdy Solution

STEP 1

What is this asking? We're figuring out how much money we'll have after 4 years if we invest $25,000\$25,000 at a 7%7\% interest rate, both monthly and continuously. Watch out! Don't mix up the formulas for monthly and continuous compounding.
They're similar, but not the same!

STEP 2

1. Monthly Compounding
2. Continuous Compounding

STEP 3

We'll use the formula for monthly compound interest: A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}.
This formula is awesome because it lets us see how our money grows over time with regular compounding.

STEP 4

Here, AA is the **future value** (what we want to find!), PP is the **principal** ($25,000\$25,000), rr is the **annual interest rate** (0.070.07), nn is the **number of times interest is compounded per year** (1212 for monthly), and tt is the **number of years** (44).

STEP 5

Let's plug those values into our formula: A=25000(1+0.0712)124A = 25000(1 + \frac{0.07}{12})^{12 \cdot 4}.

STEP 6

Now, we just need to crunch the numbers! A=25000(1+0.0712)4825000(1.005833)4825000(1.323095)33077.38A = 25000(1 + \frac{0.07}{12})^{48} \approx 25000(1.005833)^{48} \approx 25000(1.323095) \approx 33077.38.
So, with monthly compounding, our investment will be worth approximately $33,077.38\$33,077.38 after 4 years.

STEP 7

Now, for continuous compounding, we'll use a slightly different formula: A=PertA = Pe^{rt}.
This formula models the growth of our investment when interest is compounded constantly.

STEP 8

Here, AA is still the **future value**, PP is still the **principal** ($25,000\$25,000), rr is still the **annual interest rate** (0.070.07), and tt is still the **number of years** (44). ee is just the exponential constant.

STEP 9

Let's plug the values into our formula: A=25000e0.074A = 25000e^{0.07 \cdot 4}.

STEP 10

Time to calculate! A=25000e0.2825000(1.32313)33078.25A = 25000e^{0.28} \approx 25000(1.32313) \approx 33078.25.
With continuous compounding, our investment will be worth approximately $33,078.25\$33,078.25 after 4 years.

STEP 11

a. With monthly compounding, the accumulated value is approximately $33,077.38\$33,077.38. b. With continuous compounding, the accumulated value is approximately $33,078.25\$33,078.25.

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