Question

You are a student in ECON 110, and your test is 24 hours from now. You can spend this time either studying or sleeping. Suppose you know that the impact of studying and sleeping on the test score is ?(?, ?) = 8?0.5 + ?, where ? is the number of hours you spend studying (“working”) and ? is the number of hours you spend sleeping (“resting”). Assume your goal is to maximize your score on the test. a) How many hours will you study? b) How many hours will you sleep? c) What is your grade on the test if you make the optimal choices above (Hint: what is ?(?∗, ?∗))?

Answer 1

The score will be maximized when the differentiation of the score function with respect to number of hours studying will be zero. Also, it is given that w+r=24 (because we have 24 hours till the test and we can either spend them on study or sleep). In other words

g(w,r)=8w^.5+24-w

Differntiating it, we get

dg(w,r)/dw=4/w^.5-1=0

w^.5=4

w=16.

So, we will maximize our score if we spend 16 hours studying (answer to a) and 8 hours sleeping (answer to b).

At this level, our score would be

g(16,8)=8*16^.5+8

=8*4+8

=32+8

=40 (answer to c).

Our score would be 40.