Video: Solving Exponential Equations with Continuously Compounded Interest

Benjamin invested $2800 at 3.2% interest compounded continuously. After how many years and months will the value of his investment be doubled?

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Video Transcript

Benjamin invested 2800 dollars at 3.2 percent interest compounded continuously. After how many years and months will the value of his investment be doubled?

So, in this question, what we’re looking at is interest that’s compounded continuously. So, we have a formula for continuous compound interest. And that is 𝐴 is equal to 𝑃𝑒 to the power of 𝑟𝑡. And this is where 𝐴 is the amount of the end of the investment. 𝑃 is the principal amount. Then we have 𝑟 which is the interest rate in decimal. And then 𝑡 which is years.

So, if we take a look at our question, we know that the amount at the end that we’re looking for is 2800 multiplied by two. And that’s because we want to know after how many years and months the value of the investment will be doubled. So, this is gonna give us a value of 5600. 𝑃 is gonna be 2800 because that was our initial investment, or our principal amount. 𝑟 is going to be equal to 0.032. That’s cause our interest rate was 3.2 percent.

And we get that because 3.2 percent is 3.2 over 100 because percent means out of 100. Well, if you divided 3.2 by 100, then it means each of our digits is going to go two place values to the right. And we’re left with 0.032. And then finally, 𝑡 we don’t know cause 𝑡 is what we want to find out. Because we want to find out after how many years and months the value of the investment is going to be doubled.

So, when we substitute in our values, we get 5600 is equal to 2800 multiplied by 𝑒 to the power of 0.032𝑡. So therefore, next what we do is we divide each side of the equation by 2800. And we get two is equal to 𝑒 to the power of 0.032𝑡. So, as the value we’re looking for, so 𝑡, is in the exponent, what we want to do now is deal with logs. And because we’re dealing with 𝑒, what we’re gonna do is we’re gonna take the natural logarithm of each side.

And when we do that, what we get is ln two is equal to ln 𝑒 to the power of 0.032𝑡. We’re using the relationship we know. We know that if we take the natural logarithm of 𝑒 to the power of 𝑥, then it’s just gonna be equal to 𝑥. So therefore, we get ln two is equal to 0.032𝑡. So then, if we divide by 0.032, we’re gonna get ln two over 0.032 is equal to 𝑡.

So then, when we calculate this, we get 𝑡 is equal to 21.66084939. And this is in years. But we want it in years and months. So, what we need to do now, well, we know it’s gonna be 21 years. But we need to work out what 0.6608 et cetera is in months.

Well, to do that, we’ll multiply that by 12, which, when we do that, is approximately equal to 7.9. So therefore, we can say that Benjamin’s investment will be doubled after 21 years and eight months. And we’ve rounded it to eight months because at 21 years and seven months it wouldn’t have doubled. However, at 21 years and eight months, it would have doubled.

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