Answer The initial investment is P ≈ $ 6376 _ The time to double is t ≈ 15.4 y e a r s _ Explanation Given information : Annual % rate is 4.5%&Amount after 10 years is $10000.00 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 {r=0.045, A=10000.00, t = 10 years} 10000 = P e 0.045 ⋅ 10 Use symmetric property of equality which states that if a = b then b = a to rewrite the equation P e 0.045 ⋅ 10 = 10000 Divide by e 0.045 ⋅ 10 on both sides P e 0.045 ⋅ 10 e 0.045 ⋅ 10 = 10000 e 0.045 ⋅ 10 Simplify fraction on both sides of the equation P ≈ 6376 Calculation (Continued): Description Steps Replace 6376 for P &0.045for r in the equation A = P e r t A = 6376 e 0.045 t To find the time to double, we need to replace 12752 for A {2 times 6376} and set up an equation 12752 = 6376 e 0.045 t Use symmetric property of equality which states that if a = b then b = a to rewrite the equation 6376 e 0.045 t = 12752 Divide by 6376 on both sides 6376 e 0.045 t 6376 = 12752 6376 Simplify fraction on both sides of the equation e 0.045 t = 2 Take natural logarithm on both sides ln e 0.045 t = ln ( 2 ) Apply the logarithmic rule ln e X = X in left side of the equation 0.045 t = ln 2 Divide by 0.045 on both sides 0.045 t 0.045 = ln 2 0.045 Simplify fraction on both sides of the equation t ≈ 15.40 y e a r s Conclusion: Initial Investment Annual % Rate Time to Double Amount After 10 years $6376 4.5 % ▶Given) 15.4 years $10000(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 11E

To determine

To find : the Initial investment and time to double.

Expert Solution & Answer

Answer to Problem 11E

The initial investment is P≈$6376_

The time to double is t≈15.4 years_

Explanation of Solution

Given information : Annual % rate is 4.5%&Amount after 10 years is $10000.00

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 {r=0.045, A=10000.00, t = 10 years}	10000=Pe0.045⋅10
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	Pe0.045⋅10=10000
Divide by e0.045⋅10 on both sides	Pe0.045⋅10e0.045⋅10=10000e0.045⋅10
Simplify fraction on both sides of the equation	P≈6376

Calculation (Continued):

Description	Steps
Replace 6376 for P &0.045for r in the equation A = Pert	A =6376e0.045t
To find the time to double, we need to replace 12752 for A {2 times 6376} and set up an equation	12752 =6376e0.045t
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	6376e0.045t=12752
Divide by 6376 on both sides	6376e0.045t6376=127526376
Simplify fraction on both sides of the equation	e0.045t=2
Take natural logarithm on both sides	lne0.045t=ln(2)
Apply the logarithmic rule lneX=X in left side of the equation	0.045t=ln2
Divide by 0.045 on both sides	0.045t0.045=ln20.045
Simplify fraction on both sides of the equation	t≈15.40 years

Conclusion:

Initial Investment	Annual % Rate	Time to Double	Amount After 10 years
$6376	4.5%▶Given)	15.4 years	$10000(Given)

Chapter 3.5, Problem 10E

Chapter 3.5, Problem 12E

Chapter 3 Solutions

EBK PRECALCULUS W/LIMITS

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01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY

Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY

Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY