Determine the time necessary for P dollars to double when it is invested at interest rate r compounded annually, monthly, daily, and continuously. (Round your answers to two decimal places.) r = 13%

For an interest of 13%, the amount of time it will take the principal to double is:

Annually: About 5.67 years.

Monthly: About 5.36 years.

Daily: About 5.33 years.

Continuously: About 5.33 years.

Step-by-step explanation:

The standard formula for compound interest is given by:

Where P is the principal, A is the amount afterwards, r is the interest rate, n is the amount of times it is compounded per year, and t is the number of years.

We want to find t for n=1 (annually), n=12 (monthly), n=365 (daily) and continuously when our final amount is double our principal.

In other words, A=2P. Therefore:

We can divide both sides by P:

Since our rate is 13%, r=0.13. Hence:

We now have our general formula which we can use to solve.

Annually:

For an annual compound, n=1. Therefore:

Simplify :

We can take the log (base 10) of both sides:

We can move the t in front:

Therefore:

So, for an annual compound, it will take about 5.67 years for the principal to double.

Monthly:

For monthly compounds, n=12. Therefore:

We can take the log of both sides:

Again, we can move the 12t to the front:

Therefore:

So, for monthly compounds, it will take about 5.36 years for the principal to double.

Daily:

For daily compounds, n=365. Therefore:

Take the log of both sides and moving the 365t to the front yields:

Hence:

So, for daily compounds, it will take about 5.33 years for the principal to double.

Continuously:

For continuous compound, we will need to use the continuous compound formula:

So, A=2P:

Dividing both sides by P:

Since r=13% or 0.13:

This time, we can take the natural log of both sides:

So:

The natural log of e is simply 1. Therefore:

Hence:

For, for continuously compounded, it will take about 5.33 years for the principal to double.

hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb