Chapter 9.6, Problem 49E

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

Investments If an IRA is a variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of $1000 is S = 1000 [ 1 + 0.01 r 12 ] 240 What is the rate of change of S with respect to r, and what does it tell us if the interest rate is(a) 6 % ? (b) 12 % ? (a) To determine To calculate: The rate of change of S with respect to r. Interpret the rate if the interest rate is 6%. If an IRA is variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of$1000 is S=1000[1+0.01r12]240.

Explanation

Given Information:

The future value S that accumulates from an initial investment of $1000 is: S=1000[1+0.01r12]240 Where, IRA is variable-rate investment for 20 years at rate r percent, compounded monthly. Formula used: The power rule of differentiation, ddxxn=nxnâˆ’1 Calculation: Consider the provided function, S=1000[1+0.01r12]240 Consider (1+0.01r12) to be u, S=1000u240 Differentiate both sides with respect to r, dSdr=ddr(1000u240) Simplify using the power rule, dSdr=ddr(1000u240)=1000â‹…240â‹…u240âˆ’1dudr=240,000u239dudr Substitute (1+0.01r12) for u, dSdr=240,000(1+0.01r12)239ddr(1+0.01r12)=240,000(1+0 (b) To determine To calculate: The rate of change of S with respect to r. Interpret the rate if the interest rate is 12%. If an IRA is variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of$1000 is S=1000[1+0.01r12]240.

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