Question

Dale has 2000 dollars to invest. He has a goal to have 5800 in this invest ment in 9 years. At what annual rate compounded continuously will Dale reach his goal?

Answer 1

Annual rate of 12.56%

Explanation:

To find the annual rate, we need to use the compound interest formula:

M = Mo * (1+r)^t

where M is the goal value (M = 5800), Mo is the inicial value (Mo = 2000), r is the annual rate we want to find, and t is the number of years (t = 9). Then we can calculate the equation to find the value of r:

5800 = 2000 * (1+r)^9

(1+r)^9 = 5800/2000 = 2.9

1+r = 1.1256

r = 0.1256 = 12.56%

Answer 2

Dale will reach his goal at an annual rate of 11.83%.

Explanation:

The formula for continuos compounding is given by:

`A(t) = Pe^(rt)`

In which A is the amount after t years, P is the principal(initial amount) and r is the annual rate.

Dale has 2000 dollars to invest.

This means that
`P = 2000`

He has a goal to have 5800 in this invest ment in 9 years.

So
`A(9) = 5800`

At what annual rate compounded continuously will Dale reach his goal?

This is r.

`A(t) = Pe^(rt)`

`5800 = 2000e^(9r)`

`e^(9r) = (58)/(20)`

`e^(9r) = 2.9`

`\ln{e^(9r)} = ln(2.9)`

`9r = ln(2.9)`

`r = (ln(2.9))/(9)`

`r = 0.1183`

Dale will reach his goal at an annual rate of 11.83%.