TITLE: 2 variables calculus word problem
QUESTION [5 upvotes]: A certain factory using $x$ do-dads and $y$ widgets per hour will produce:
$T(x,y)=\frac{90x^2}{y}$
thingamabobs per day.
Describe the relationship between inputs  $x$ and $y$ such that $3600$ thingamabobs are produced per day. (That is, find the level curve of $T$ corresponding to $T = 3600.$)
My answer is wrong and I'm not sure why. This is what I did.
$$3600=90x^2y$$
$$\frac{3600}{90x^2}=y$$
$$\frac{40}{x^2}= y$$
Where did I go wrong in answering this question? What should the right solution be?

REPLY [3 votes]: Your error is in the very first line.  We have
$$
T(x, y) = \frac{90x^2}{y}
$$
and we want to set $T(x, y) = 3600$, but you wrote
$$
3600 = 90x^2y
$$

If we instead begin
$$
3600 = \frac{90x^2}{y}
$$
then we can multiply both sides by $\frac{y}{3600}$ to get
$$
y = \frac{x^2}{40}
$$
which is the reciprocal of your expression.  This kind of makes sense because your original set-up had $y$ where you should have had $\frac{1}{y}$.
There are, of course, other ways to express this basic relationship, all fundamentally equivalent.