# Video: AP Calculus AB Exam 1 β’ Section I β’ Part B β’ Question 90

Michelle made a \$16000 investment. The Investment is supposed to grow at the rate of 460π^(0.2π‘) dollars per year, where π‘ is measured in years. What is then the approximated amount Michelle will have in the bank after 5 years?

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### Video Transcript

Michelle made a 16000-dollar investment. The Investment is supposed to grow at the rate of 460π to the 0.2π‘ dollars per year, where π‘ is measured in years. What is then the approximated amount Michelle will have in the bank after five years?

Letβs start by labelling the amount Michelle has in the bank as π΄. In fact, we could call this π΄ of π‘, such that π΄ is a function of π‘. Therefore, π΄ of π‘ is the amount in the bank after π‘ years. Now, the question tells us that Michelle makes an initial investment of 16000 dollars. This means that at π‘ is equal to zero, π΄ is equal to 16000 dollars. Alternatively, we could write π΄ of nought is equal to 16000 dollars. The next piece of information given in the question, which we can use, is that the investment is supposed to grow at a rate of 460π to the 0.2π‘ dollars per year. This is the rate at which the amount grows. So we can say that dπ΄ by dπ‘ is equal to 460π to the 0.2π‘.

Now the question is asking us to find how much Michelle will have in the bank after five years. And so, therefore, what weβre trying to find is π΄ of five. In order to find π΄ of five, we first need to find π΄ of π‘. We could do this by integrating dπ΄ by dπ‘ with respect to π‘. When integrating 460π to the 0.2π‘, we first notice that 460 is a constant. And so we can factor it out of the integral. Now, what we need to integrate is π to the 0.2π‘ with respect to π‘. Here, we have a function within a function. In order to integrate this, we can use the reverse chain rule.

We find the differential of the function within the function, which is 0.2π‘. The differential of 0.2π‘ with respect to π‘ is simply 0.2. Therefore, in order to integrate π to the 0.2π‘, we start by integrating the exponential part as usual, which simply gives us each π to the 0.2π‘. And then, we mustnβt forget to divide by the differential of 0.2π‘, which is 0.2. This is equivalent to multiplying by one over 0.2.

Now that we have completed our indefinite integration, we mustnβt forget to add a constant of integration, which we can call π. Simplifying our function, we find that π΄ of π‘ is equal to 2300π to the 0.2π‘ plus π. Now, we can use our initial condition which is that π΄ of naught is equal to 16000. We simply substitute π΄ is equal to 16000 and π‘ is equal to zero into this equation. Now, we can simplify this and we obtain that π is equal to 13700.

For our next step, weβll simply substitute the value of π back into π΄ of π‘. Now, we have found an equation for π΄ of π‘ which is π΄ of π‘ is equal to 2300π to the 0.2π‘ plus 13700. Since weβre trying to find π΄ of five, our final step here will be to substitute π‘ is equal to five into π΄. This gives us a value of 19952 dollars and five cents. Rounding this to the nearest dollar gives us the amount in Michelleβs account after five years will be 19952 dollars.