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

Question

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take $50,000 to triple in value at an annual rate of 7.5% compounded continuously?

Accepted Answer

Hello. Today we're going to solve a word problem Using continuous compounding interest. So let's just go ahead and recall what that equation is. So continuous compounding interest is P equal to P initial times E raised to r times T. P is going to be your value with respect to time respect to time. P initial is going to be your original values. R. Is going to be your rate of change and T. Is going to equal to time. So we're going to use this to solve this problem. So let's just go ahead and write down the equation. P is equal to P initial times E raised to the R times T. Now the problem says that we are going to figure out how long it will take for a change to happen. So since we're figuring out how long for a change to happen we want to solve for T. In order to solve for T. We can divide both sides by P initial. So P over P initial is going to equal to E raise the R. T. Now we are told that we have 30,000 counts of a certain bacteria. That's just a population size which we don't need for this problem. And we want this population to be 100 times in value. So 100 times in value. That means that we want our values to be changed by 100 times. So P over P initial is going to be equal to 100 and we are also told that the rate of change is 10.5%/h. While that means that our is going equal to 10.5%. Now we need to change this percentage into a decimal in order to do that. We're gonna go ahead and divide 10.5 by 100. Doing so is going to give me a rate of 0.105. Now what we're gonna do is we're gonna take these two values and plug them into our equation to solve for T. Doing so is going to give me 100 is equal to erase the power of 0.105 times t. Now in order to solve for T, I'm gonna need to bring it down from this exponent and in order to do that we can take the natural log of both sides of this equation. Doing so is going to give me the natural log of equal to the natural log of E. race to the power of 0.105 T. Okay now what natural log allows me to do is take this exponent and bring it to the front of this natural log as a product. So doing so is going to give me the natural log of 100 equal to 0. T. Times the natural log of E. And recall that the natural log of E is equal to one. So we're gonna go ahead and replace the natural log of E with one. Doing so is going to give me the natural log of equal to 0.105 times t. And finally, to solve for T. We're gonna divide both sides by 0.105. So T. is going to equal to the natural log of Over 0.105. And plugging this entire value into a calculator is going to give us T approximately equal to 43.86. And this is going to be our solution. That also means that the solution to this problem is going to be C. So I hope this video helps you in understanding how to use continuous compounding interest and I'll go ahead and see you all in the next video.

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take $50,000 to triple in value at an annual rate of 7.5% compounded continuously?

Key Concepts

Continuous Compounding

Exponential Growth

Natural Logarithm

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