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September 13th, 2017, 01:59 PM  #1 
Newbie Joined: Sep 2017 From: New York, NY Posts: 1 Thanks: 0  Linear Programming Problem!
The semester has just started and you want to come up with an optimal daily schedule for your offclass time, i.e. you want to know the time you need to allocate to studying, and the time you need to allocate for extra curricular activities. For a more balanced life, you want to ensure that the time allocated to studying is at least 20% of your time, and the time for extracurriculars is at least 20% of your time. Furthermore, you cannot allocate more than 3 hours to any of those two. If the maximum amount of time you can spend daily is 5 hours, and you believe that studying is 1.5 times more beneficial to you than extracurriculars, find the daily schedule that maximizes your benefit. I really only need the constraints, but if you want to answer the whole thing feel free (: thank you! 
September 13th, 2017, 02:45 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,011 Thanks: 1044 
total time is $T=5hr$ $T_{study} \geq (0.2)T = 1hr$ $T_{ec} \geq (0.2)T = 1hr$ $1 \leq T_{study} \leq 3$ $1 \leq T_{ec} \leq 3$ $benefit = (1.5)T_{study} + T_{ec}$ It seems pretty clear that you want to maximize the study time subject to the constraint that you spend at least an hour on extracurricular activities. However the other constraint kicks in and you can only spend a maximum of $3hr$ studying. So just by inspection the optimal solution is $T_{study}=3,~T_{ec}=2$ 

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