of a rectangle that has the given perimeter and a maximum area.
Perimeter : 80 meters
Im still not sure how to set these up to prove with calculus.
2x+2y=80?
Let a rectangle have length and width .
Then by the first condition, we have .
Let . It follows from above that .
Substituting this into the area equation, we have .
Since we want to maximize area, it follows that . Now, we find the critical point:
(To verify its a max, we differentiate again to get )
Since , it follows that .
Therefore, the dimensions of the rectangle that gives us the maximum area actually form a square.
Does this make sense?