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Among all rectangles that have a perimeter of 60, find the dimensions of th one whose area is largest.

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2 Answers

Among all rectangles that have a perimeter of 60, find the dimensions of the one whose area is largest.
 
Here's an algebraic solution.
 
x = width of rectangle
 
y = length of rectangle
 
x + y = 30 ==> y = 30 – x
 
Area, A = xy = x(30–x)
 
A(x) is a quadratic that graphs as a parabola opening down (-x^2);
so Vertex will be maximum point on parabola.
 
Zeros of A(x): x = 0 and x = 30
 
Axis of Symmetry: x = 30/2 = 15.
 
Vertex: (15,A(15))
 
So largest rectangle with perimeter of 60 has:
 
width = 15, and
 
length = 30 – 15 = 15.
 
It's a square.

Comments

I see where you are going with that and it is a great solution for someone who has the math background to understand it, but it seems to me to be making the problem harder than it needs to be.
You made an intuitive leap to x+Y=30 but aren't taking intuition any further.  You have 2 equations, a quadratic, zeros, vertex, axis of symmetry, a parabola, and maxima of the parabola.  This takes at least the completion of Algebra 1 and maybe more to both understand the vocabulary (zeros, vertex, symmetry, maxima) and solve it.  So we would be talking about a high school student.
 
It seems to me that anyone who knows/understands how to multiply, find area, and what a square is should be able to solve this by intuition.  Maybe 6th grade.  I would much rather show someone how to understand and use their intuition. Why do it the hard way?
If the perimeter is 60 then the special rectangle (a square) has sides of 15 each.
 << Yes, a square is a rectangle - it is a special case (or kind) of rectangle.>>
 
To get the area we multiply the length and width - i.e. 15 x 15 or 15 squared.
 
If you know about the square function, you understand squared number get larger very quickly.
Much more quickly than just multiplying two numbers.
So test it - what are the sides of a rectangle that isn't a square?
What about the rectangle that has two sides 14 - what are the other two sides?
Multiply that out and see if it is bigger or smaller than 15 squared.