Math  /  Algebra

QuestionFormula: A(T)=Pe(RT)A(T)=\mathrm{Pe}^{\wedge}(R T) \#6: If \1500isinvestedataninterestrateof1500 is invested at an interest rate of 4.5 \%$ per year, compounded continuously, find the value of the investment afte 14 years. (4 Points)

Studdy Solution

STEP 1

What is this asking? How much money will we have if we invest \$1500 at a 4.5% interest rate, compounded continuously, for 14 years? Watch out! Don't forget to convert the percentage to a decimal!
Also, make sure to use the correct formula for continuous compounding.

STEP 2

1. Define the variables
2. Calculate the final value

STEP 3

Alright, let's break this down!
We're given the formula A(T)=PeRTA(T) = P e^{RT}.
This looks intimidating, but it's just a fancy way of calculating how much money we'll have after a certain amount of time.

STEP 4

A(T)A(T) is the **amount** we're trying to find – that's our goal! PP is the **principal**, which is the **initial investment** of $1500\$1500.
So, we have P=1500P = 1500.

STEP 5

RR is the **interest rate**.
It's given as 4.5%, but we need to write it as a decimal.
To do this, we divide 4.5 by 100, which gives us R=0.045R = 0.045.

STEP 6

TT is the **time** in years, which is given as **14 years**.
So, T=14T = 14.

STEP 7

Now, let's plug everything into our formula!
We have A(14)=1500e0.04514A(14) = 1500 \cdot e^{0.045 \cdot 14}.

STEP 8

First, let's calculate the exponent: 0.04514=0.630.045 \cdot 14 = 0.63.
So, our formula becomes A(14)=1500e0.63A(14) = 1500 \cdot e^{0.63}.

STEP 9

Now, we need to calculate e0.63e^{0.63}.
Using a calculator, we find that e0.631.8776e^{0.63} \approx 1.8776.

STEP 10

Finally, we multiply this by our principal: 15001.87762816.41500 \cdot 1.8776 \approx 2816.4.

STEP 11

After 14 years, the investment will be worth approximately $2816.40\$2816.40.

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