Dean has $20$ days and $4500$ acres in which to plant corn and soybeans

Question

I was solving this problem:

Dean has $20$ days in which to plant corn and soybeans. The corn can be planted at a rate of $250$ acres per day and the soybeans at a rate of $200$ acres per day. He has $4500$ acres available for planting these two crops. If the profit is $\$26$ per acre on corn and $\$30$ per acre on soybeans, how much of each should Dean plant? What is the maximum profit?

And I couldn't figure it out, so I just googled it and these inequalities came out;

X = acres of soy Y = acres of corn Then, the constraints are: Y/250 + X/200 <=20

X+Y <= 4500

X>=0, Y >=0

I get all of them, but what does Y/250 + X/200 <=20 represent? I thought the inequality has to be something like

250y+200x <= 4500

x+y<=20

Why am I wrong, and why is this Y/250 + X/200 <=20 correct?

Answer 1

You are not wrong. Both formulations are correct, but the meaning of $x$ and $y$ is different between the two.

In the Google solution $X$ and $Y$ represent acres, so, for example, $X/200$ has units of acres / (acres/day) = days, and the constraint $Y/250 + X/200 \le20$ is the constraint that Dean has 20 days to plant.

In your solution $x$ and $y$ represent days, so $x+y\le 20$ is the constraint that Dean has 20 days to plant.