   Chapter 6.5, Problem 21E

Chapter
Section
Textbook Problem

# (a) If $3000 is invested at 5 % interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously.(b) If A ( t ) is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by A ( t ) . To determine (a): To calculate: The amount at the end if the interest is compounded i. annually ii. semi-annually iii. monthly iv. weekly v. daily, and vi. continuously Explanation Given: P =$3000, r = 5% per annum and t = 5 years.

Let P = Principal amount, A = amount at the end, r = rate of interest per annum and t = time (in years).

Formula used:

1. If the interest is compounded n-times in a year then the amount at the end is

A=P(1+r100n)nt

2. If the interest is compounded continuously then the amount at the end is

A(t)=Per100t

Calculation:

(i) If the interest is compounded annually:

In this case the interest is compounded one time in a year, therefore, the amount at the end is:

A=P(1+r100n)nt    =3000(1+5100)5    =3828.84

Therefore, the amount at the end is $3828.84 if the interest is compounded annually. (ii) If the interest is compounded semi-annually: In this case the interest is compounded 2 times in a year, therefore, the amount at the end is: A=P(1+r100n)nt =3000(1+52×100)5×2 =3840.25 Therefore, the amount at the end is$3840.25 if the interest is compounded semi-annually

To determine

(b)

To write:

The differential equation with initial condition for A(t)

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