Answer The initial investment is P ≈ $ 1122 _ The Annual % Rate is r = 5.78 % _ Explanation Given information : Time to double is 12 years&Amount after 10 years is $2000.00 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 We are given time to double is 12 years, so let us substitute A=2P in equation A = P e r t 2 P = P e 12 r Use symmetric property of equality which states that if a = b then b = a to rewrite the equation P e 12 r = 2 P In the process of solving for r divide by P on both sides P e 12 r P = 2 P P Simplify fraction on both sides of the equation e 12 r = 2 Take natural logarithm on both sides ln e 12 r = ln ( 2 ) Apply the logarithmic rule ln e X = X in left side of the equation 12 r = ln 2 Divide by 12 on both sides 12 r 12 = ln 2 12 Calculation (Continued): Description Steps Simplify fraction on both sides of the equation r ≈ 0.0578 Rewrite the decimal as percentage r ≈ 0.0578 ⋅ 100 % = 5.78 % Substitute the given information in the formula A = P e r t {r=0.0578, A=2000.00, t = 10 years} 2000 = P e 0.0578 ⋅ 10 Use symmetric property of equality which states that if a = b then b = a to rewrite the equation P e 0.0578 ⋅ 10 = 2000 Divide by e 0.0578 ⋅ 10 on both sides P e 0.0578 ⋅ 10 e 0.0578 ⋅ 10 = 2000 e 0.0578 ⋅ 10 Simplify fraction on both sides of the equation P ≈ 1122 Conclusion: Initial Investment Annual % Rate Time to Double Amount After 10 years $1122 5.78 % 12 years (Given) $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 12E

To determine

To find : the Initial investment and Annual % Rate.

Expert Solution & Answer

Answer to Problem 12E

The initial investment is P≈$1122_

The Annual % Rate is r=5.78%_

Explanation of Solution

Given information : Time to double is 12 years&Amount after 10 years is $2000.00

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
We are given time to double is 12 years, so let us substitute A=2P in equation A = Pert	2P =Pe12r
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	Pe12r=2P
In the process of solving for r divide by P on both sides	Pe12rP=2PP
Simplify fraction on both sides of the equation	e12r=2
Take natural logarithm on both sides	lne12r=ln(2)
Apply the logarithmic rule lneX=X in left side of the equation	12r=ln2
Divide by 12 on both sides	12r12=ln212

Calculation (Continued):

Description	Steps
Simplify fraction on both sides of the equation	r≈0.0578
Rewrite the decimal as percentage	r≈0.0578⋅100% =5.78%
Substitute the given information in the formula A = Pert {r=0.0578, A=2000.00, t = 10 years}	2000=Pe0.0578⋅10
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	Pe0.0578⋅10=2000
Divide by e0.0578⋅10 on both sides	Pe0.0578⋅10e0.0578⋅10=2000e0.0578⋅10
Simplify fraction on both sides of the equation	P≈1122

Conclusion:

Initial Investment	Annual % Rate	Time to Double	Amount After 10 years
$1122	5.78%	12 years (Given)	$10000(Given)

Chapter 3.5, Problem 11E

Chapter 3.5, Problem 13E

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