Answer The initial investment is P ≈ $ 303581 _ Explanation Given information : Amount available for retirement is $500,000, rate of interest is 5% and duration of investment is 10 years 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 ( 1 + r n ) n t Calculation: Description Steps Substitute the given information in the formula A = P ( 1 + r n ) n t {A=500000, r = 0.05, t = 10, & n = 12 because compounded monthly} 500000 = P ( 1 + 0.05 12 ) 12 ⋅ 10 Use symmetric property of equality which states that if a = b then b = a to rewrite the equation P ( 1 + 0.05 12 ) 12 ⋅ 10 = 500000 Simplifying the left side of the equation P ( 1.0041 6 ¯ ) 120 = 500000 Calculation (Continued): Description Steps Divide by ( 1.0041 6 ¯ ) 120 on both sides P ( 1.0041 6 ¯ ) 120 ( 1.0041 6 ¯ ) 120 = 500000 ( 1.0041 6 ¯ ) 120 Simplify fraction on both sides P = $ 303581 Conclusion: The principal $303581 should be invested at rate of 5%, compounded monthly, so that $500,000 will be available for retirement in 10 years

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Differential Equations

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Question

Chapter 3.5, Problem 13E

To determine

To find : the principal P.

Expert Solution & Answer

Answer to Problem 13E

The initial investment is P≈$303581_

Explanation of Solution

Given information : Amount available for retirement is $500,000, rate of interest is 5% and duration of investment is 10 years

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(1+rn)nt

Calculation:

Description	Steps
Substitute the given information in the formula A = P(1+rn)nt {A=500000, r = 0.05, t = 10, & n = 12 because compounded monthly}	500000 =P(1+0.0512)12⋅10
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	P(1+ 0.05 12)12⋅10=500000
Simplifying the left side of the equation	P(1.00416¯)120=500000

Calculation (Continued):

Description	Steps
Divide by (1.00416¯)120 on both sides	P( 1.0041 6 ¯ )120( 1.0041 6 ¯ )120=500000( 1.0041 6 ¯ )120
Simplify fraction on both sides	P=$303581

Conclusion:

The principal $303581 should be invested at rate of 5%, compounded monthly, so that $500,000 will be available for retirement in 10 years

Chapter 3.5, Problem 12E

Chapter 3.5, Problem 14E

Chapter 3 Solutions

EBK PRECALCULUS W/LIMITS

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