Alice has only $24$ hours to study for an exam, and without preparation she will get $200$ points out of $1000$ points on the exam. It is estimated that her exam score will improve by $x(50−x)$ points if she reads her lecture notes for $x$ hours and $y(48−y)$ points if she solves review problems for y hours, but due to fatigue she will lose $(x+y)^2$ points. What is the maximum exam score she can obtain? Round your answer to the nearest integer.
I got $y=23/2$ and $x=25/2$.
When I plugged those back in I got $468.75$, and then I need to add the $200$ points. So my answer would be $668.75$ which would round to $669$. But this is not correct. What am I doing wrong?
A few things:
First of all (and most importantly), you call this a Lagrange problem (and your solution does satisfy x + y = 24), but is it really? Seems improbable that the optimal strategy would involve no sleep. I'd look in the interior, $x + y < 24$.
Second, I don't get the same numbers you get. I get the same x and y values (using Lagrange, though these are not optimal), but not the same test score. Are we using the same score function? Mine is: $$ 200+ x(50-x)+y(48-y)-(x+y)^2$$ Using the values you get from Lagrange, I see that as 512.5 . Am I missing something?