Answer The annual percentage rate is r ≈ 11.02 % _ The time to double is t ≈ 6.29 y e a r s _ Explanation Given information : Initial Investment is $500 &Amount after 10 years is $1505.00 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 and annual interest rate r (in decimal form) is given by the formula: A = P e r t Logarithmic property: ln e X = X Calculation: Description Steps Substitute the given information in the formula A = P e r t {P=500, A=1505.00, t = 10 years} 1505 = 500 e 10 r Use symmetric property of equality which states that if a = b then b = a to rewrite the equation 500 e 10 r = 1505 Divide by 500 on both sides 500 e 10 r 500 = 1505 500 Simplify fraction on both sides of the equation e 10 r = 3.01 Take natural logarithm on both sides ln e 10 r = ln 3.01 Calculation (Continued): Description Steps Apply the logarithmic rule ln e X = X in left side of the equation 10 r = ln 3.01 Divide by 10 on both sides 10 r 10 = ln 3.01 10 Simplify fraction on both sides of the equation r ≈ 0.1102 Rewrite the decimal as percentage r = 0.1102 ⋅ 100 % = 11.02 % Replace 500 for P &0.1102 for r in the equation **emsp; A = P e r t A = 500 e 0.1102 t To find the time to double, we need to replace 1000 for A and set up an equation 1000 = 500 e 0.1102 t Use symmetric property of equality which states that if a = b then b = a to rewrite the equation 500 e 0.1102 t = 1000 Divide by 500 on both sides 500 e 0.1102 t 500 = 1000 500 Simplify fraction on both sides of the equation e 0.1102 t = 2 Take natural logarithm on both sides ln e 0.1102 t = ln ( 2 ) Apply the logarithmic rule ln e X = X in left side of the equation 0.1102 t = ln 2 Divide by 0.1102 on both sides 0.1102 t 0.1102 = ln 2 0.1102 Simplify fraction on both sides of the equation t ≈ 6.29 y e a r s Conclusion: Initial Investment Annual % Rate Time to Double Amount After 10 years $500 (Given) 11.02 % 6.29 years $1505.00(Given)

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Differential Equations

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Question

Chapter 3.5, Problem 10E

To determine

To find : the Annual % Rate and time to double.

Expert Solution & Answer

Answer to Problem 10E

The annual percentage rate is r≈11.02% _

The time to double is t≈6.29 years_

Explanation of Solution

Given information : Initial Investment is $500 &Amount after 10 years is $1505.00

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 and annual interest rate r (in decimal form) is given by the formula: A = Pert

Logarithmic property: lneX=X

Calculation:

Description	Steps
Substitute the given information in the formula A = Pert{P=500, A=1505.00, t = 10 years}	1505=500e10r
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	500e10r=1505
Divide by 500 on both sides	500e10r500=1505500
Simplify fraction on both sides of the equation	e10r=3.01
Take natural logarithm on both sides	lne10r=ln3.01

Calculation (Continued):

Description	Steps
Apply the logarithmic rule lneX=X in left side of the equation	10r=ln3.01
Divide by 10 on both sides	10r10=ln3.0110
Simplify fraction on both sides of the equation	r≈0.1102
Rewrite the decimal as percentage	r=0.1102⋅100% =11.02%
Replace 500 for P &0.1102 for r in the equation **emsp;A = Pert	A =500e0.1102t
To find the time to double, we need to replace 1000 for A and set up an equation	1000 =500e0.1102t
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	500e0.1102t=1000
Divide by 500 on both sides	500e0.1102t500=1000500
Simplify fraction on both sides of the equation	e0.1102t=2
Take natural logarithm on both sides	lne0.1102t=ln(2)
Apply the logarithmic rule lneX=X in left side of the equation	0.1102t=ln2
Divide by 0.1102 on both sides	0.1102t0.1102=ln20.1102
Simplify fraction on both sides of the equation	t≈6.29 years

Conclusion:

Initial Investment	Annual % Rate	Time to Double	Amount After 10 years
$500 (Given)	11.02%	6.29 years	$1505.00(Given)

Chapter 3.5, Problem 9E

Chapter 3.5, Problem 11E

Chapter 3 Solutions

EBK PRECALCULUS W/LIMITS

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