A researcher has funds to buy enough computing power to number-crunch a problem

Question

A researcher has funds to buy enough computing power to number-crunch a problem in 9 years if she buys now. However, computing power per dollar doubles every 13 months. When should she buy her computers to have finished the problem as soon as possible? Give your answer as a decimal in months, accurate within 0.1 months. Suppose the problem would take c months on current computers. What is the largest value of c for which she should buy the computers immediately? Give an answer as a decimal accurate to 0.1 months.

Explanation / Answer

let t = time in months from now when researcher purchases computing power.
let T = time to complete problem in months

9 years= 108 months

T(t)=t+117(1/2)-t/13

differenciating wrt t and equating to 0 we get

1- 117 (t/13) (1/2)(-t-13)/13  =0

1- 117t/13 * (1/2)(-t-13)/13 =0

b) T(t) = t + C*2(-t/13)

dT(t)/dt=0

1+ C (-t/17) 2(-t-13)/13 =0

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