Chapter 4, Problem 81RE

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Compound Interest A deposit of $400 is made in an account that earns interest at an annual rate of 2.5%. How long will it take for the balance to double when the interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously? (a) To determine To calculate: The time required so that the deposited amount will become double if the amount of$300 is deposited in an account at the interest rate of 2.5% and the interest is compounded annually.

Explanation

Given Information:

The amount of $300 is deposited in an account at the interest rate of 2.5% and the interest is compounded annually. Formula used: The formula to compute the balance amount after t years when interest is compounded n times per year is, A=P(1+rn)nt Where, P is the deposited amount, A is the amount after t years, r is the interest rate in decimals, t is the number of years and n is the number of times compounded per year. The property of logarithms for the expression lnxn is lnxn=nlnx where x and y are real numbers greater than 0. Calculation: Consider that the amount of$300 is deposited in an account at the interest rate of 2.5% and the interest is compounded annually.

Here, P=$300, r=2.5%. The number of times compounded per year is 1 as the interest is compounded annually, that is, n=1. As the amount will become double of the deposited amount after t years, then the amount after t years is 2P, that is, A=2P=2($300)=$600 Simplify the rate as, r=2.5%=2.5100=0.025 Substitute 0.025 for interest rate,$300 for the amount deposited, 1 for number of times compounded per year and $600 for the amount after t years in the formula, “A=P(1+rn)nt” as,$600=$300(1+0 (b) To determine To calculate: The time required so that the deposited amount will become double if the amount of$300 is deposited in an account at the interest rate of 2.5% and the interest is compounded monthly.

(c)

To determine

To calculate: The time required so that the deposited amount will become double if the amount of $300 is deposited in an account at the interest rate of 2.5% and the interest is compounded daily. (d) To determine To calculate: The time required so that the deposited amount will become double if the amount of$300 is deposited in an account at the interest rate of 2.5% and the interest is compounded continuously.

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