Chapter 12.5, Problem 42E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Investing When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account. If $20,000 is invested (when t = 0) and the amount in the account after 22 years is$280,264, find the function that gives the value of the investment as a function of t. What is the interest rate on this investment?

To determine

To calculate: The investment function after time t, and also interest rate of the investment if the account rate is dPdt=kP and $20,000 is invested and the amount in the account after 22 years is$280,264.

Explanation

Given Information:

The investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the amount in the account.

dPdt=kP

Where P is in dollars, t is in the years, and k is a constant.

If $20,000 is invested and the amount in the account after 22 years is$280,264.

Formula used:

Separable equation is of the form:

g(y)dy=f(x)dx

Integration on both the sides yields the solution of this equation.

Calculation:

Consider the investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the amount in the account.

dPdt=kP

Where P is in dollars, t is in the years, and k is a constant.

If $20,000 is invested and the amount in the account after 22 years is$280,264.

Firstly, separate the above differential equation as:

dPP=kdt

Integrate the provided function as:

dPP=kdtlnP=kt+CP=ekt+CP=Cekt

At the given condition t=0,P=20,000 put in the above equation as:

20,000=Cee(0)C=20,000

At the given condition t=22,P=280,264 put in the above equation as:

280,264=20,000e22t14

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