I have a question regarding using calculus to solve an optimisation problem which is quite wordy. It is as follows:
A researcher has funds to buy enough computing power for 7 years. Computing power per dollar doubles every 22 months.
a) When should he buy his computers to have the problem finished as soon as possible?
b) Suppose the problem takes c months on current computers. What is the largest value of c for which he should buy the computers immediately.
If possible I was hoping to have some help being walked through the process of just formulating this type of problem and techniques to solving it as my only experience with these questions as so far I have only worked on questions about maximising the areas of boxes or cylinders where certain constraints are usually given rather than having to formulate the problem first then find the solution.
So far I believe my best guess is the following:
Cost(c) = 22c^2 where c is less than or equal to 7 and then somehow I'd have to start using the first and second derivative tests?
Thank you,
Michael
My attempt at part (a):
We know that if you purchase the computers now, it will take 7 years (84 months) to finish the problem. Every 22 months, the time it takes for the computer to run the program halves. The time it will take to finish the problem would equal the sum of the time bought and the amount of time it takes to solve at time t.
So we can model this situation by $T(t) = t + 84 (1/2)^{t/22}$, where t is in months, and T(t) is the time it takes to finish the problem.
At this point, you can take the derivative and set it equal to 0, solve for t, and this should find you the optimal time to purchase the computers.