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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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Finance You are investing P dollars at an annual interest rate of r, compounded continuously, for t years. Which of the following options would you choose to get the highest value of the investment? Explain your reasoning.

(a) Double the amount you invest.

(b) Double your interest rate.

(c) Double the number of years.

To determine

The best choice from the following options to obtain the highest balance amount and give reason when an amount of P dollars is invested for t years at the interest rate of r which is compounded continuously. The options are as follows:

(a). Double the invested amount

(b). Double the interest rate

(c). Double the number of years

Explanation

Given Information:

An amount of P dollars is invested for t years at the interest rate of r which is compounded continuously. The options are as follows:

(a). Double the invested amount

(b). Double the interest rate

(c). Double the number of years

Consider that the amount of P dollars is invested for t years at the interest rate of r which 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.

The balance amount after t years when the amount of P dollars is invested at the interest rate of r is,

A=Pert

Assume x=rt, then

A=Pex ……(1)

Now, consider the part (a) to calculate the balance amount when the invested amount gets doubled,

As the invested amount gets doubled, then the invested amount is,

P=2P

The balance amount A1 when the invested amount gets doubled is as follows,

A1=Pex=2Pex

From equation (1), the balance amount A1 is,

A1=2A

Thus, the balance amount will get double when the invested amount gets doubled.

Now, consider the part (b) to calculate the balance amount when the interest rate gets doubled,

As the interest rate gets doubled, then the new interest rate is,

r=2r

The balance amount A2 when the interest rate gets doubled is as follows,

A2=Pert=Pe2rt

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