Optimization: Show that, of all the rectangles with a given

[SCGirl]

Show that, of all the rectangles with a given area, the one with the smallest perimeter is a square, and show that of all the rectangles with a given perimeter, the one with the greatest area is a square.

Please help. I don't even know where to start.

 

[stapel]

One starting point might be to draw a rectangle, label the sides with variables indicating length and width, and plug these into the "area" formula.

Pick some constant for the area, and plug the variables and the constant into the "area" formula. Solve for one of the variables. Then plug the other variable and the expression (with the one variable expressed in terms of the other variable and the constant) into the "perimeter" equation.

Differentiate, and find the minimizing dimensions. Show that the minimal perimeter occurs when the length equals the width.

The other part works similarly.

If you get stuck, please reply clearly showing (or describing) what you have done. Thank you.

Eliz.

 

[SCGirl]

Thanks. It was not hard to show at all!