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(a) How long will it take an investment to double in valueif the interest rate is 6$\%$ compounded continuously?(b) What is the equivalent annual interest rate?

a) $$=11.55~yr$$b) $$=6.185 \%$$

Calculus 1 / AB

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

Derivatives

Differentiation

Applications of the Derivative

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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guys, welcome back. This problem is an interest from we'll use our compound ing interest equation, which is a It's legal to pee e to the R T where peas our initial value is how many get him, How many we have in the account at the very end, ours are interest rate per year, and t is our time and years. So we want to figure out how long it's gonna take to double. We're doubling equation. Then where you can say that a equals two times p two times we initially put into the account. And when you're has a re of 0.6 because it is 6% per year 6% so are Siegel. 2.6 I got into our equation. So we get to pee Eagle two p times E point, Europe six t Peace will cancel. We take the natural longer both sides with the natural log of to natural Aga and the you will cancel out and we're left with 0.6 t we have our time is equal to natural log of to over 0.6 connected this value of 11 0.55 years. So we can say that it will take 11.55 years for the investment to double with a 6% annual compound interest rate. Next, I want to find an equivalent annual interest rate. We'll use our equation a The p I'm the one plus R over end times end of the tea and we know that's still a has to be equal to two p. You know that. Are you feeling year by year? For one year So and it's just one. How many years go by? How many times a year in Triscuits compound it. We know we have a totally years of 11.55 No, I want to solve for R so we have to pee. You go to be one plus are over one times one times 11.55 He's once again will cancel. We have to is eagle toe one plus R times erased 11.55 Now we can raise both. Besides, to be won over 11.55 on the right inside. Those will cancel out. We're left with two. You want over 11.55 The one plus r. We are to the 1/11 0.55 minus one. And then we did a calculation. You get a value of point Europe six 0.6 95 but it makes more sense for this answer and then percentage even our value of 6.185%. So in part A, we found how much time it would take for the investment double and value at an interest rate of 6% per year. Compound continuously. And then in Part B, we found an equivalent annual interest rate. That is, if we compound once per year for a time of 11.5 years, or 11.55 years and double the principal. But with interest rate be, we found that interest rate would be 6.185%. Next includes both parts of the problem. Thanks for watching

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