Use the formula for continuous compounding to solve Exercises 84–...

Question

Use the formula for continuous compounding to solve Exercises 84–85. What annual rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

Accepted Answer

Hello. Today we're gonna be solving for a rate of change using the equation of continuous compounding interest. So before we jump into this let's just go ahead and recall what the equation for continuous compounding interest is. So that equation is going to be P equal to P initial times E. Raised to the power of our times T. P. Is going to be your value with respect to time respect to time P initial is going to be your original value. R. Is going to be your rate or interest rate and T. Is going to be time. So we're going to use this to figure out what our interest rate is. So the problem states that we want to find an annual rate that is required for an investment subject with continuous compounding interest to be quadrupled in seven years. So let's just start by writing down our equation. P. Is equal to P initial times E. Raised to the R. T. And again we are asked to find an annual rate. So our goal is to go ahead and solve for R. What we can do first is go ahead and divide both sides of this equation by P initial. And that's going to give me P over P initial equal to E. Raised to the R. Times T. Now we are told that the values want to be quadrupled so we can allow P over P initial to be four. We don't care about what the initial values are. We just know that those values need to be quadrupled. So that's why we're setting that equal to four. And we are told that we want this to be quadrupled in seven years. So we're going to allow t time to equal to seven. So what we're gonna do with these values is not plug in to our equation. Doing so is going to give me four equal to erase the power of seven R. Now what we need to do is go ahead and bring this seven are down from the exponent in order to do that. We can go ahead and take the natural logarithms of both sides of the equation. Doing so is going to give me the natural log of four equal to the natural log of the race of the power of seven are now again this natural log function allows me to bring the exponent to the front of the natural log as a product. Doing so is going to give me the natural log of four equal to seven R. Times the natural log of E. And let's just go ahead and recall that the natural log of E is equal to one. So replacing the natural log of E with one is going to give me the natural log of four equal to seven R. Times one, which is just going to be seven R. And in order to solve for R. I'm just gonna go ahead and divide both sides by seven. That is going to give me are equal to the natural log of 4/7. Now what I can do from here is go ahead and evaluate the natural log of 4/7 by putting it into a calculator. So that's going to give me an approximate value of 0.1980. Now again, we want to go ahead and convert this To a percentage in order to do that. We're gonna multiply this value by 100 and that is going to give me R is equal to 19.80%. And rounding this to the nearest% 20%. And that is going to be the solution to this problem. That also means that the solution to this problem is going to be be so I hope this video helps you in understanding how to use the equation for continuous compounding interest. And I'll go ahead and see you all in the next video.

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