Quadratic models in factor form

Question

width x
length 36-4x
Max area x(36-4x)

What is the width if we maximize the room. Calculate the width to maximize the room.

Answer 1

Area = x(36-4x) = -4x² + 36x

 

This is a quadratic equation whose graph is a parabola.  Since the coefficient of the x² term is negative (-4), it's an inverted parabola with the vertex at the top.  The vertex is thus the maximum point, representing the maximum room area.  The vertex is always located at the point:

 

x = -b/2a

 

where a is the coefficient of the x² term and b is the coefficient of the x term.  In this case, a = -4 and b = 36, so:

 

x = -36/(2)(-4) = 36/8 = 9/2

 

So the room dimensions that yield maximum area are:

 

Width = x = 9/2 = 4 ¹/₂

Length = 36-4x = 36-4(9/2) = 18