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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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Think About It Are the times required for the investments in Exercises 73 and 74 to quadruple twice as long as the times for them to double? Give a reason for your answer and verify your answer algebraically.

To determine

Whether the time required for the investment to quadruple is twice the time required for the investment to double or not and give reason for the answer. Also verify the answer for the investment of $3000 at the interest rate of 8.5% and 5.5% which is compounded continuously algebraically.

Explanation

Given Information:

The amount of $3000 is invested at the interest rate of 8.5% and 5.5% which is compounded continuously.

Consider that the deposited amount P (in dollars) will become double at the interest rate of r when the interest is compounded continuously.

The formula to compute the balance amount after t years when interest is compounded continuously is,

A=Pert

Where, P is the deposited amount, A is the amount after t years, r is the interest rate in decimals and t is the number of years.

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.

Substitute r for interest rate, P for the amount deposited and 2P for the amount after t years in the formula, “A=Pert” as,

2P=Pert

Divide both sides by P as,

2PP=PertPert=2

Take natural logarithm on both sides as,

ln(ert)=ln2

The inverse property of logarithms for the expression lnex is lnex=x.

Now, apply the inverse property of lnex as,

rt=ln2

Divide both sides by r as,

rtr=ln2rt=ln2r

So, the time required for the investment to double is ln2r.

Now, consider that the deposited amount P (in dollars) will become quadruple at the interest rate of r when the interest is compounded continuously.

As the amount will become quadruple of the deposited amount after t years, then the amount after t years is 4P, that is A=4P.

Substitute r for interest rate, P for the amount deposited and 4P for the amount after t years in the formula, “A=Pert” as,

4P=Pert

Divide both sides by P as,

4PP=PertPert=4

Take natural logarithm on both sides as,

ln(ert)=ln4ln(ert)=ln(2)2

The property of logarithms for the expression lnxn is lnxn=nlnx where x is real number greater than 0.

Now, apply the property of lnex and ln(xn) as,

rt=2ln2

Divide both sides by r as,

rtr=2ln2rt=2ln2r

So, the time required for the investment to quadruple is 2ln2r.

From the above evaluation, it is seen that the time required for the investment to quadruple is twice the time required for the investment to double.

Thus, the time required for the investment to quadruple is twice the time required for the investment to double.

Now, consider that the amount of $3000 is deposited in an account at the interest rate of 8.5% and the interest is compounded continuously.

Here P=$3000, r=8.5%.

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($3000)=$6000

Simplify the rate as,

r=8.5%=8.5100=0.085

Substitute 0.085 for interest rate, $3000 for the amount deposited and $6000 for the amount after t years in the formula, “A=Pert” as,

$6000=$3000e0.085t

Divide both sides by $3000 as,

$6000$3000=$3000e0.085t$3000e0.085t=2

Take natural logarithm on both sides as,

ln(e0.085t)=ln2

Now, apply the inverse property of lnex as,

0.085t=ln2

Divide both sides by 0.085 as,

0.085t0.085=ln20.085t0.6930.0858.15 years

So, the deposited amount will be doubled in approximately 8.15 years when the interest rate is 8.5%.

Now, as the amount will become quadruple of the deposited amount after t years, then the amount after t years is 4P, that is,

A=4P=4($3000)=$12000

Substitute 0

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