A bank offers an account with a 5% annual interest rate, compounded continuously. Approximately how long would it take for a $5,000 investment to double?

9514 1404 393

  13.9 years

Step-by-step explanation:

The doubling time in years for an account earning interest continuously is given exactly by the formula ...

  t = 69.31472/r . . . . where r is the annual interest rate in percent

We have r = 5, so the doubling time is ...

  t = 69.31472/5 = 13.8629 . . . years

It will take approximately 13.9 years for the investment to double in value.

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Additional comment

The account multiplier for continuous interest is ...

  e^(rt)

That multiplier is 2 when ...

  2 = e^(rt)

  ln(2) = rt . . . take the natural log of both sides

  t = ln(2)/r = 0.6931472/r

Multiplying by 100 so r is in percent, this is ...

  t = 69.31472/r . . . . with r in percent

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This compares with the "rule of 72" that is used to estimate doubling time when interest is paid periodically. That rule would tell you doubling time at 5% is approximately 72/5 = 14.4 years. The value "72" actually depends on interest rate and frequency of compounding, so that rule only gives a crude estimate.

hello from mrbilldoesmath!

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

discussion:

in general, the factorization of a^3 + b^3 is

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

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes

343x^3 + 216y^3 =

((7x + 6y) (   (7x)^2 - 7x*6y + (6y)^2)   =

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

you,

mrb

(a + b) (a^2 - a b + b^2)

given the equation is 8+9z=-6, the first step would be to subtract the 8 from both sides:

8+9z-8=-6-8

since 8 minus 8 = 0, this would cancel out the 8. and since -6 minus 8 equals -14, the equation would look like this

9z = -14