Chapter 3.4, Problem 71E

(a)

To determine

To find : the time necessary for P dollars to double.

Answer to Problem 71E

The time necessary for P dollars to double is $\boxed{\underset{\_}{t\approx 27.73years}}$

Explanation of Solution

Given information : Amount invested $P=\$2500$ , interest rate $r=0.025$ , and it 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
Substitute the given information in the formula   $A=P{e}^{rt}${   $P=2500$, $A=5000$because amount doubling, r = 0.025}	$5000=2500e^{0.025\cdot t}$
Use symmetric property of equality which states that if $a=b$ , then $b=a$ to rewrite the equation	$2500e^{0.025\cdot t}=5000$
Dividing 2500 on both sides of the equation	$\frac{2500e^{0.025\cdot t}}{2500}=\frac{5000}{2500}$
Simplify fraction in both sides of the equation	$e^{0.025\cdot t}=2$

Calculation (Continued):

Description	Steps
Take natural logarithm on both sides of the equation	$\ln e^{0.025\cdot t}=\ln 2$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.025t=\ln 2$
Divide by 0.025 on both sides	$\frac{0.025t}{0.025}=\frac{\ln 2}{0.025}$
Simplify fraction on both sides of the equation	$t\approx 27.726years$

Conclusion:

It would take time of 27.73years for $2500 dollars to double when it is invested at interest rate $r=0.025$ compounded continuously.

(b)

To determine

To find : the time necessary for P dollars to triple.

Answer to Problem 71E

The time necessary for P dollars to triple is $\boxed{\underset{\_}{t\approx 43.94years}}$

Explanation of Solution

Given information : Amount invested $P=\$2500$ , interest rate $r=0.025$ , and it 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
Substitute the given information in the formula   $A=P{e}^{rt}${ $P=2500$,   $A=7500$because amount tripling, r = 0.025}	$7500=2500e^{0.025\cdot t}$
Use symmetric property of equality which states that if $a=b$ , then $b=a$ to rewrite the equation	$2500e^{0.025\cdot t}=7500$
Dividing 2500 on both sides of the equation	$\frac{2500e^{0.025\cdot t}}{2500}=\frac{7500}{2500}$
Simplify fraction in both sides of the equation	$e^{0.025\cdot t}=3$

Calculation (Continued):

Description	Steps
Take natural logarithm on both sides of the equation	$\ln e^{0.025\cdot t}=\ln 3$
Apply the logarithmic rule $\ln e^X=X$ in left side of the equation	$0.025t=\ln 3$
Divide by 0.025 on both sides	$\frac{0.025t}{0.025}=\frac{\ln 3}{0.025}$
Simplify fraction on both sides of the equation	$t\approx 43.94years$

Conclusion:

It would take time of 43.94 years for $2500 dollars to triple when it is invested at interest rate $r=0.025$ compounded continuously.