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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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A deposit of $500 is made in an account that earns interest at an annual rate of 4%. 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 $500 is deposited in an account at the interest rate of 4% and the interest is compounded annually.

Explanation

Given Information:

The amount of $500 is deposited in an account at the interest rate of 4% 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 $500 is deposited in an account at the interest rate of 4% and the interest is compounded annually.

Here, P=$500, r=4%.

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($500)=$1000

Simplify the rate as,

r=4%=4100=0.04

Substitute 0.04 for interest rate, $500 for the amount deposited, 1 for a number of times compounded per year and $1000 for the amount after t years in the formula, A=P(1+rn)nt as,

$1000=$500(1+0

(b)

To determine

To calculate: The time required so that the deposited amount will become double if the amount of $500 is deposited in an account at the interest rate of 4% 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 $500 is deposited in an account at the interest rate of 4% 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 $500 is deposited in an account at the interest rate of 4% and the interest is compounded continuously.

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