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Two algorithms A and B solve the same algorithmic problem, A taking n^3 seconds and B taking n days.
(i) Which algorithm is asymptotically preferable?   
(ii) How large does n need to be before B takes one-quarter of the time taken by A?

How do I go about solving these?

My answer for (i) is that B is preferable as n grows at a faster rate asymptotically. Days and seconds here count as a constant and therefore do not matter as n approaches infinity.

for ii) my guess is 2 days. But wondered if others got the same

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I could be completely wrong here, but I think you want this

24*60*60*n = n^3 * 1/4  

which when plugged into wolphram alpha gives

587.87....

or

-587.87 in some alternate universe ;)

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This looks like a brute force type of algorithm would work to give you a better feal for the problem. I would take a look at the total time it takes for the two functions to be equal and what happens above and below that threshold. Also, for good practice it may be good to look for multiple solutions.

y=n^3 seconds
y=n * (60 seconds * 60 minutes * 24 hours) seconds = n seconds per day 

Then look through incrementing values of n for points where the value of y is equal between the two functions.