Question

Can you help me with a word problem?Under continuous compounding, the amount of time t in years required for an investment to double is a function of the interest rate r according to the formula:=tln2r(a)If you invest $3000 how long will it take the investment to reach $6000 if the interest rate is 2.5%? Round to one decimal place.(b)If you invest $3000 how long will it take the investment to reach $6000 if the interest rate is 8%? Round to one decimal place.(c)Using the doubling time found in part (b), how long would it take a $3000 investment to reach $12,000 if the interest rate is 8%? Round to one decimal place.

Answer 1

a) the amount of time in years it will take for the given investment to double is;

`t=27.7\text{ years}`

b) the amount of time in years it will take for the given investment to double is;

`t=8.7\text{ years}`

Step-by-step explanation:

Given that under continuous compounding, the amount of time t in years required for an investment to double is a function of the interest rate r according to the formula;

`t=(\ln 2)/(r)`

a) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 2.5%.

`r=2.5\text{\%=}\frac{\text{2.5}}{100}=0.025`

Applying the given formula;

`\begin{gathered} t=(\ln 2)/(0.025) \\ t=27.7\text{ years} \end{gathered}`

Therefore, the amount of time in years it will take for the given investment to double is;

`t=27.7\text{ years}`

b) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 8%.

`r=8\text{ \%}=(8)/(100)=0.08`

Applying the given formula;

`\begin{gathered} t=(\ln 2)/(0.08) \\ t=8.7\text{ years} \end{gathered}`

Therefore, the amount of time in years it will take for the given investment to double is;

`t=8.7\text{ years}`