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

Answer:

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;

$t=8.7\text{ years}$

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;

$t=\frac{\ln 2}{r}$

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;

$\begin{gathered} t=\frac{\ln 2}{0.025} \\ t=27.7\text{ years} \end{gathered}$

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%.

$r=8\text{ \%}=\frac{8}{100}=0.08$

Applying the given formula;

$\begin{gathered} t=\frac{\ln 2}{0.08} \\ t=8.7\text{ years} \end{gathered}$

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

$t=8.7\text{ years}$