Question

The amount of money in an account with continuously compounded interest is given by the formula A= Pe^rt, where P is the principl, R is the annual interest rate, and t is time in years.
Calculate to the nearest tenth of a year how long it take for an amount of a year how long it takes for an amount of money to double if interest is compounded continuously at 7.5%
A. 8.2 years
B. 9.2 years
C. 0.9 years
D. 0.6 years

Answer 1

ima go with c but not 100%

Answer 2

you can solve the equation
2P = P* e^(rt) , plug in r = .075

Answer 3

2p= Pe^(0.5)(t) ?

Answer 4

almost

Answer 5

2P= Pe^(0.075*t)

Answer 6

ohhh. sorry

Answer 7

no problem :)

Answer 8

now you can divide both sides by P, that will eliminate P

Answer 9

why is P on both sides?

Answer 10

the original equation was
A = Pe ^(r * t)

Answer 11

yeah but you wrote..."2P= Pe^(0.075*t)"

Answer 12

right, because the question said, when does your principal double

Answer 13

how long will it take for an initial amount P , to double.
double P is 2P

Answer 14

ohhh okay.

Answer 15

you can also eliminate P by multiplying both sides of the equation by P

Answer 16

2=Pe(0.075*t)

Answer 17

should be
2 = e^(.075 * t )

Answer 18

okay. got it.

Answer 19

do i just plug in now?

Answer 20

yeah thats correct

Answer 21

I'd say the first thing to do would be to convert the interest rate to a CONTINUOUSLY compounded rate.
The formula for that is:
http://www.1728.org/rate.htm
Continuous rate = (e^annual rate) -1
Continuous rate = (2.718281828 ^ .075) -1
Continuous rate = 0.0778841509

Since we need to calculate the time for doubling, we'll say the initial amount is $100 and the ending amount is $200.
The compounding formula solved for years is:
http://www.1728.org/compint2.htm
Years = {log(total) -log(Principal)} / log(1 + rate)
Years = {log(200) -log(100)} / log(1+ 0.0778841509)
Years = (2.3010299957 - 2) / 0.0325720861
Years = .3010299957 / 0.0325720861
Years = 9.2419624207

Looks like it is "b" don't you think?

Answer 22

There is a simpler way (but NOT as "dead on" exact) to calculate this.
The APPROXIMATE method to calculate the "doubling time" of an interest rate is called the "Rule of 72".
You get the number 72 and divide it by the interest rate.
Doubling time in years = 72/annual rate
Doubling time in years = 72/7.5
Doubling time in years = 9.6

We can easily see that "b" is the only answer that is even remotely close to 9.6.
Of course if you require dead on accuracy, then the previous method is the one to use.

Answer 23

Upon further consideration, we must consider what the calendar tells us ... and it says:

"Merry Christmas"

Answer 24

I actually remember that now!

Answer 25

Glad I could help out. (Luckily Santa helped me as he dropped by my house.) :-)