$2000 is invested at 5% interest compounded continuously. How much… Step by Step

Step by step

The question

Question: $2000 is invested at 5% interest compounded continuously.

How much is in the account after 10 years?

1 Read the numbers

P is the money put in: $2000.

P = $2000: the money put in

r is the rate.

5% is 5/100: 0.05.

r = 5% = 0.05

t is the time in years: 10.

t = 10 years

With continuous interest, the amount is A = P times e to the power rt.

e is the special number from the ln lesson, about 2.718.

A = Pe^rt

e ≈ 2.718

Careful!

The rate goes in as a decimal: 0.05, not 5.

Careful: not r = 5 ✗

2 Put them in

We put the numbers in.

A = 2000 × e^{0.05 × 10}

3 Work it out

The power first: 0.05 times 10 is 0.5.

0.05 × 10 = 0.5

So A is 2000 times e to the power 0.5.

A = 2000 × e^0.5

With a calculator, e to the power 0.5 is about 1.64872.

e^0.5 ≈ 1.64872

Now times 2000.

A ≈ 2000 × 1.64872

3297.4425...

Money is rounded to cents: about $3297.44.

A ≈ $3297.44

4 Check

Let's check by going back: 3297.44 divided by 1.64872 is about $2000, the money put in.

$3297.44 ÷ 1.64872 ≈ $2000

And it makes sense.

Simple interest would give $3000.

Interest on interest gives a bit more.

So the answer is right!

simple interest: $2000 + $1000 = $3000 < $3297.44

Let's look back.

First: read the numbers.

Then: put them in.

Then: work it out.

Last: check.

Answer