Global Extrema/Area of a Rectangle

[Quantum_Grid]

Homework Statement

Find the dimensions of of the rectangle with perimeter 200 meters that has the largest area.

Homework Equations

The Attempt at a Solution

This is in the section on Global Maxima/Minima so I know it has to be something with graphing a formula and finding the maxima, but I cannot figure where to start on this. The book is no help (Applied Calculus - Hughes-Hallett, et.al) and this is not the first time it has presented problems that it does not even began to describe in the text.

Can someone tell me where to start?

 

[Dunkle]

What is the equation for the perimeter of a rectangle with length x and height y (and what do you have to set this equal to)? What is the equation for the area of this rectangle?

 

[Quantum_Grid]

What is the equation for the perimeter of a rectangle with length x and height y (and what do you have to set this equal to)? What is the equation for the area of this rectangle?

Ok, so I have 200=2(x+y), and the area is A=xy, right? I must be missing something terribly obvious, because I am still stuck on where to go from here, except that maybe 100=x+y. How does this fit into a Global Extrema problem?

 

[Dunkle]

Okay, good! Now, you need to write the area in terms of just one variable so that you can find the maximum of the area function. Any ideas?

 

[Quantum_Grid]

y=A/x? Man, I can almost get this, I feel like I'm so close, but where does the perimeter function come in, y=100-x?

Is this problem an an easy one? Because it seems like it, and it's making me feel so dumb that I can't figure it out!

 

[Dunkle]

You are so close! You are trying to maximize the area, so you need to write the area as a function of x (or y) only. You can solve for x in terms of y (and vice versa) using the perimeter constraint. 2x+2y = 200 => y = 100-x => A = x(100-x). Now how do we find the value of x when this function is at its maximum? Once you find this x, you can solve for y using the relationship we found between x and y using the perimeter constraint. Is this clear?

 

[Quantum_Grid]

50x50! Dude, thanks for your patience on that one, it was driving me crazy! It makes sense now.

 

[Dunkle]

You're welcome! Also, note that this answer may have been what you expected from the beginning: a square!