Chapter 3.8, Problem 21E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 1 for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(r). (a) To determine To find: The amounts due to the end of 5 years if the interest compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, (vi) continuously Explanation Given: Initial amount is$3000.

Interest rate is 5%.

Number of years is 5.

Formula used:

The value of the investment is A(t)=A0(1+rn)nt.

Where,

A0 is the initial investment.

r is the interest rate.

n is the number of times the interest is compounded per year.

t is the number of years.

Calculation:

Section (i)

If n=1, the amount of A(5) is computed as follows,

Substitute A0=3000,r=0.05 and t=5 in A(t)=A0(1+rn)nt

A(5)=(3000)(1+0.051)1(5)=(3000)(1+0.05)5=(3000)(1.05)5=3828.84

Thus, the amount is A(5)=3828.84.

Section (ii)

If n=2, the amount of A(5) is computed as follows,

Substitute A0=3000,r=0.05 and t=5 in A(t)=A0(1+rn)nt,

A(5)=(3000)(1+0.052)2(5)=(3000)(1+0.025)10=(3000)(1.025)10=3840.25

Thus, the amount is A(5)=3840.25.

Section (iii)

If n=12, the amount of A(5) is computed as follows,

Substitute A0=3000,r=0.05 and t=5 in A(t)=A0(1+rn)nt,

A(5)=(3000)(1+0.0512)12(5)=(3000)(1+0

(b)

To determine

To write: The differential equation and initial condition satisfied by A(t) .

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