Question

how much money has to be invested at 2.3% interest compounded continuously to have $41,000 after 17 years

Answer 1

How were you taught to do this?

Answer 2

im not sure how to do this.

Answer 3

Very well ^^
When interest is compounded *continuously*, it follows this formula:
Suppose we start with the principal (or initial) amount, P.

And we have the interest rate, r, compounded continuously for a number of years, t.

Then the final amount, A, is given by the following formula:

\[\huge A = P e^{rt}\]
Where e is the constant approximately 2.71828

Answer 4

What is the answer?

Answer 5

Why don't we find out?
Here, we're asked for the amount to start with, that is, the principal, so P is what we're looking for.

What is the final amount, A?

Answer 6

im not sure because i do not get the formula at all. please help.

Answer 7

I'm explaining the formula now ^^
My question right now has nothing to do with the formula, my question is:

What is the amount we want to end up with?

Answer 8

$41,000 after 17 years ..

Here are the answer choices..

A. $27,731.59
B. $27,741.97
C. $27,762.66
D. $27,793.53

Answer 9

Never mind the answer choices for now.
So the amount we want to end up with, A, this is 41,000
And the time is 17 years.
So let's substitute accordingly...
\[\huge \color{blue}{41000} = P e^{r(\color{red}{17})}\]
Now what about the continuously compounded rate of interest, r?
What is it?

Answer 10

2.3%

Answer 11

Which is 0.023
\[\huge {41000} = P e^{\color{green}{0.023}({17})}\]

Answer 12

Now, I hope you have your calculator handy, because you'll need it...
How do you get P alone on one side?
As in... how do you get rid of everything else but P on the right-side of the equation?

Answer 13

I DONT UNDERSTAND THIS AT ALL

Answer 14

No need to shout ^^
Suppose we have a simple
\[\Large x = yz\]
If we wanted to get the y alone on the right-side of the equation, all we would need to do is divide both sides of the equation by z, giving us...

\[\Large \frac x z = \frac {yz}z\]\[\Large \frac xz = \frac{y\cancel{z}}{\cancel{z}}\]\[\Large \frac xz = y\]

Answer 15

\[\huge {41000} = P e^{{0.023}({17})}\]
Similarly, here, all we need to do is divide both sides of the equation by the bit on the right side that ISN'T P. That is to say, \(\Large e^{0.023(17)}\)
Thus giving us:
\[\huge \frac{41000}{e^{0.023(17)}} = \frac{P e^{{0.023}({17})}}{e^{0.023(17)}}\]
\[\huge \frac{41000}{e^{0.023(17)}} = \frac{P \cancel{e^{{0.023}({17})}}}{\cancel{e^{0.023(17)}}}\]

\[\huge \frac{41000}{e^{0.023(17)}} = P\]

That wasn't so hard, was it? ^^

Answer 16

I still dont understand because none of the answer choices match the answer on my calculator..

Answer 17

Nevermind. Thank you so much <3 !

Answer 18

You were very helpful :)