Chapter 3.5, Problem 17E

(a)

To determine

To find : the time necessary for P dollars to triple when investment compounded continuously.

Answer to Problem 17E

The completed table for time t (in years) necessary for P dollars to triple when it is invested at an interest rate r compounded continuously is given below

r	$2\%$	$4\%$	$6\%$	$8\%$	$10\%$	$12\%$
t (in years)	54.93	27.47	18.31	13.73	10.99	9.16

Explanation of Solution

Given information : Amount invested is P dollars, annual rate of interest are 2%, 4%, 6%, 8%, 10% &12%compounded continuously

Concept Involved:

Solving for a variable means getting the variable alone in one side of the equation by undoing whatever operation is done to it.

Formula Used:

For continuous compounding, after t years, the balance A in an account with principal P, number of times interest applied per time period n and annual interest rate r (in decimal form) is given by the formula: $A=P{e}^{rt}$

Logarithmic property: $\ln e^X=X$

Calculation:

Description	Steps
For rate of interest $r=2\%$ substitute the given information in the continuous compounding formula $A=P{e}^{rt}$ {A=3P because amount doubling, r = 0.02}	$3P=P{e}^{0.02\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.02\cdot t}=3P$

Calculation (Continued):

Description	Steps
Divide by $P$on both sides	$P{e}^{0.02\cdot t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.02\cdot t}=3$
Take natural logarithm on both sides	$\ln e^{0.02\cdot t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.02t=\ln 3$
Divide by 0.02 on both sides	$0.02t/0.02=\ln 3/0.02$
Simplify fraction on both sides of the equation	$t\approx 54.93years$
It would take time of 54.93 years for P dollars to triple when it is invested at interest rate $r=2\%$ compounded continuously.
For rate of interest $r=4\%$substitute the given information in the continuous compounding formula $A=P{e}^{rt}${A=3P because amount doubling, r = 0.04}	$3P=P{e}^{0.04\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.04\cdot t}=3P$
Divide by $P$on both sides	$P{e}^{0.04t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.04\cdot t}=3$
Take natural logarithm on both sides	$\ln e^{0.04\cdot t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.04t=\ln 3$
Divide by 0.04 on both sides	$0.04t/0.04=\ln 3/0.04$
Simplify fraction on both sides of the equation	$t\approx 27.47years$
It would take time of 27.47 years for P dollars to triple when it is invested at interest rate $r=4\%$ compounded continuously.
For rate of interest $r=6\%$substitute the given information in the continuous compounding formula $A=P{e}^{rt}${A=3P because amount doubling, r = 0.06}	$3P=P{e}^{0.06\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.06\cdot t}=3P$
Divide by $P$on both sides	$P{e}^{0.06t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.06\cdot t}=3$
Take natural logarithm on both sides	$\ln e^{0.06\cdot t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.06t=\ln 3$
Divide by 0.06 on both sides	$0.06t/0.06=\ln 3/0.06$
Simplify fraction on both sides of the equation	$t\approx 18.31years$
It would take time of 18.31 years for P dollars to triple when it is invested at interest rate $r=6\%$ compounded continuously.
For rate of interest $r=8\%$substitute the given information in the continuous compounding formula $A=P{e}^{rt}${A=3P because amount doubling, r = 0.08}	$3P=P{e}^{0.08\cdot t}$

Calculation (Continued):

Description	Steps
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.08\cdot t}=3P$
Divide by $P$on both sides	$P{e}^{0.08t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.08\cdot t}=3$
Take natural logarithm on both sides	$\ln e^{0.08\cdot t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.08t=\ln 3$
Divide by 0.08 on both sides	$0.08t/0.08=\ln 3/0.08$
Simplify fraction on both sides of the equation	$t\approx 13.73years$
It would take time of 13.73 years for P dollars to triple when it is invested at interest rate $r=8\%$ compounded continuously.
For rate of interest $r=10\%$substitute the given information in the continuous compounding formula $A=P{e}^{rt}${A=3P because amount doubling, r = 0.1}	$3P=P{e}^{0.1t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.1t}=3P$
Divide by $P$on both sides	$P{e}^{0.1t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.1t}=3$
Take natural logarithm on both sides	$\ln e^{0.1t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.1t=\ln 3$
Divide by 0.1 on both sides	$0.1t/0.1=\ln 3/0.1$
Simplify fraction on both sides of the equation	$t\approx 10.99years$
It would take time of 10.99 years for P dollars to triple when it is invested at interest rate $r=10\%$ compounded continuously.
For rate of interest $r=12\%$substitute the given information in the continuous compounding formula $A=P{e}^{rt}${A=3P because amount doubling, r = 0.12}	$3P=P{e}^{0.12t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{e}^{0.12t}=3P$
Divide by $P$on both sides	$P{e}^{0.12t}/P=3P/P$
Simplifying fraction on both sides	$e^{0.12t}=3$
Take natural logarithm on both sides	$\ln e^{0.12t}=\ln \left(3\right)$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.12t=\ln 3$
Divide by 0.12 on both sides	$0.12t/0.12=\ln 3/0.12$
Simplify fraction on both sides of the equation	$t\approx 9.16years$
It would take time of 9.16 years for P dollars to triple when it is invested at interest rate $r=12\%$ compounded continuously.

Conclusion:

It would take time of 54.93 years for P dollars to triple when it is invested at interest rate $r=2\%$ compounded continuously.

It would take time of 27.47 years for P dollars to triple when it is invested at interest rate $r=4\%$ compounded continuously.

It would take time of 18.31 years for P dollars to triple when it is invested at interest rate $r=6\%$ compounded continuously.

It would take time of 13.73 years for P dollars to triple when it is invested at interest rate $r=8\%$ compounded continuously.

It would take time of 10.99 years for P dollars to triple when it is invested at interest rate $r=10\%$ compounded continuously.

It would take time of 9.16 years for P dollars to triple when it is invested at interest rate $r=12\%$ compounded continuously.

We can notice that as the rate increases, the time taken for the invested amount to triple will decrease.

(b)

To determine

To find : the time necessary for P dollars to triple (when investment compounded annually).

Answer to Problem 17E

The completed table for time t (in years) necessary for P dollars to triple when it is invested at an interest rate r compounded annually is given below:

r	$2\%$	$4\%$	$6\%$	$8\%$	$10\%$	$12\%$
t (in years)	55.48	28.01	18.85	14.27	11.53	9.69

Explanation of Solution

Given information : Amount invested is P dollars, annual rate of interest are 2%, 4%, 6%, 8%, 10% & 12% compounded annually

Concept Involved:

Solving for a variable means getting the variable alone in one side of the equation by undoing whatever operation is done to it.

Formula Used:

For periodic compounding, after t years, the balance A in an account with principal P, number of times interest applied per time period n and annual interest rate r (in decimal form) is given by the formula: $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Logarithmic property: $\ln m^n=n\ln m$

Calculation:

Description	Steps
When $r=2\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.02& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.02}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.02}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.02\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.02\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.02\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.02\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.02\right)=\ln 3$
Divide by ln(1.02) on both sides	$\frac{t\cdot \ln 1.02}{\ln 1.02}=\frac{\ln 3}{\ln 1.02}$
Simplify fraction on both sides of the equation gives time for r = 2%	$t\approx 55.48years$
When $r=4\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.04& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.04}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.04}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.04\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.04\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.04\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.04\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.04\right)=\ln 3$
Divide by ln(1.04) on both sides	$\frac{t\cdot \ln 1.04}{\ln 1.04}=\frac{\ln 3}{\ln 1.04}$
Simplify fraction on both sides of the equation gives time for r = 4%	$t\approx 28.01years$

Calculation (Continued):

Description	Steps
When $r=6\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.06& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.06}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.06}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.06\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.06\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.06\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.06\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.06\right)=\ln 3$
Divide by ln(1.06) on both sides	$\frac{t\cdot \ln 1.06}{\ln 1.06}=\frac{\ln 3}{\ln 1.06}$
Simplify fraction on both sides of the equation gives time for r = 6%	$t\approx 18.85years$
When $r=8\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.08& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.08}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.08}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.08\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.08\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.08\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.08\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.08\right)=\ln 3$
Divide by ln(1.08) on both sides	$\frac{t\cdot \ln 1.08}{\ln 1.08}=\frac{\ln 3}{\ln 1.08}$
Simplify fraction on both sides of the equation gives time for r = 8%	$t\approx 14.27years$

Calculation (Continued):

Description	Steps
When $r=10\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.1& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.1}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.1}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.1\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.1\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.1\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.1\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.1\right)=\ln 3$
Divide by ln(1.1) on both sides	$\frac{t\cdot \ln 1.1}{\ln 1.1}=\frac{\ln 3}{\ln 1.1}$
Simplify fraction on both sides of the equation gives time for r = 10%	$t\approx 11.53years$
When $r=12\%$, substitute the given information in the periodic compounding formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}${A=3P because amount tripling, r = 0.12& n = 1 because compounded annually}	$3P=P{\left(1+\frac{0.12}{1}\right)}^{1\cdot t}$
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	$P{\left(1+\frac{0.12}{1}\right)}^{1\cdot t}=3P$
Simplify the expression in the left side of the equation	$P{\left(1.12\right)}^t=3P$
Divide by $P$on both sides	$\frac{P{\left(1.12\right)}^t}{P}=\frac{3P}{P}$
Simplifying fraction on both sides	${\left(1.12\right)}^t=3$
Take natural logarithm on both sides	$\ln {\left(1.12\right)}^t=\ln \left(3\right)$
Apply the logarithmic rule $\ln m^n=n\ln m$ in left side of the equation	$t\cdot \ln \left(1.12\right)=\ln 3$
Divide by ln(1.12) on both sides	$\frac{t\cdot \ln 1.12}{\ln 1.12}=\frac{\ln 3}{\ln 1.12}$
Simplify fraction on both sides of the equation gives time for r =12%	$t\approx 9.69years$

Conclusion:

It would take time of 55.48 years for P dollars to triple when it is invested at interest rate $r=2\%$ compounded annually.

It would take time of 28.01 years for P dollars to triple when it is invested at interest rate $r=4\%$ compounded annually.

It would take time of 18.85 years for P dollars to triple when it is invested at interest rate $r=6\%$ compounded annually.

It would take time of 14.27 years for P dollars to triple when it is invested at interest rate $r=8\%$ compounded annually.

It would take time of 11.53 years for P dollars to triple when it is invested at interest rate $r=10\%$ compounded annually.

It would take time of 9.69 years for P dollars to triple when it is invested at interest rate $r=12\%$ compounded annually.

We can notice that as the rate increases, the time taken for the invested amount to triple will decrease.