Math  /  Algebra

QuestionQuestion 11
Suppose an investment account is opened with an initial deposit of \13,000earning13,000 earning 7.9 \%$ interest compounded continuously. How much will the account be worth after 35 years? Round to the nearest cent.
The account will be worth \ \square$ Question Help: Fritten Example Submit Question

Studdy Solution

STEP 1

What is this asking? How much money will be in the account after 35 years if we start with $13,000\$13,000 and it grows continuously at a rate of 7.9%7.9\%? Watch out! Don't forget that the interest is compounded *continuously*, so we'll need that special formula!
Also, remember to round to the nearest cent at the very end!

STEP 2

1. Define the formula
2. Plug in the values
3. Calculate the result

STEP 3

Alright, so when we have continuous compounding, we use a magical formula: A=PertA = Pe^{rt}. AA is the **final amount**, PP is the **principal** (the initial amount we put in), ee is that special **Euler's number** (approximately 2.718282.71828), rr is the **interest rate** (as a decimal!), and tt is the **time** in years.

STEP 4

Let's plug in what we know!
Our **principal**, PP, is $13,000\$13,000.
Our **rate**, rr, is 7.9%7.9\%, which we write as 0.0790.079 in decimal form.
And our **time**, tt, is **35 years**.

STEP 5

So, our formula becomes: A=13000e0.07935A = 13000 \cdot e^{0.079 \cdot 35}.

STEP 6

First, let's calculate that exponent: 0.07935=2.7650.079 \cdot 35 = 2.765.
So, we have A=13000e2.765A = 13000 \cdot e^{2.765}.

STEP 7

Now, we use a calculator to find e2.765e^{2.765}, which is approximately 15.881515.8815.
So, A=1300015.8815A = 13000 \cdot 15.8815.

STEP 8

Finally, we multiply: 1300015.8815=206459.513000 \cdot 15.8815 = 206459.5.

STEP 9

Rounding to the nearest cent, we get $206,459.50\$206,459.50.
Boom!

STEP 10

After 35 years, the account will be worth $206,459.50\$206,459.50.

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