An investment of $49,000 is made for 15 years at an interest rate of 8% compounded continuously. We need to find the value of the investment after 15 years and the total amount of interest earned.

1. Problem Description

An investment of $49,000 is made for 15 years at an interest rate of 8% compounded continuously. We need to find the value of the investment after 15 years and the total amount of interest earned.

2. Solution Steps

The formula for continuous compounding is given by:

$A = Pe^{rt}$

where:

$A$ is the amount of money accumulated after n years, including interest.

$P$ is the principal amount (the initial amount of money).

$r$ is the annual interest rate (in decimal form).

$t$ is the time the money is invested for in years.

$e$ is Euler's number (approximately 2.71828).

In this problem:

$P = 49000$

$r = 0.08$

$t = 15$

Plugging these values into the formula, we get:

$A = 49000 * e^{(0.08 * 15)}$

$A = 49000 * e^{1.2}$

$A = 49000 * 3.3201169227$

$A = 162685.72921$

Rounding to the nearest cent, we get $A = 162685.73$.

The total amount earned in compound interest is the final amount minus the principal amount:

Interest = $A - P$

Interest = $162685.73 - 49000$

Interest = $113685.73$

3. Final Answer

The value of the investment after 15 years is $162685.

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3. The total amount earned in compound interest is $113685.73.