How would Jack allocate his time to maximize his pleasure?

Question

Jack is an aspiring freshmen at a university. He realizes that “all work and no play make Jack a dull boy.” As a result, Jack wants to apportion his available time of about $10$ hours a day between work and play. He estimates that play is twice as much as fun as work. He also wants to study at least as much as he plays. However, Jack realizes that if he is going to get all his homework assignment done, he cannot play more than $4$ hours a day. How would Jack allocate his time to maximize his pleasure from both work and play?

Answer 1

Let's write $w$ for the number of hours the Jack works each day, $p$ for the number of hours that Jack plays each day, and $e$ for the amount of enjoyment that Jack gets each day.

$(1)$ Jack has $10$ hours in the day, so $$0 \leq w \leq 10, \ \ 0 \leq p \leq 10, \ \ w + p = 10.$$

$(2)$ Jacks says that play is twice as enjoyable as work, so $$e = w + 2p.$$

$(3)$ Jack says that he wants to work for at least as long as he plays, so $$p \leq w.$$

$(4)$ Jack says that he needs to work for at least $4$ hours each day, so $$4 \leq w \leq 10.$$

We can rearrange $(1)$ and substitute it into $(2)$ to get $e = p + 10$, so we'll maximize the amount of enjoyment exactly when we maximize the amount of play.

Rearranging $(3)$ and substituting it into $(1)$ gives $0 \leq p \leq 5$. Taking $(w,p) = (5,5)$ satisfies all conditions $(1)$ - $(4)$. We cannot take any higher value of $p$, and a lower one would decrease the enjoyment.

Therefore, Jack should work for $5$ hours and play for $5$ hours.