Lines, Parabola, and Systems

Question

Find two numbers whose sum is 78 and whose product is a maximum

Answer 1

Let the two numbers be x and y.  We want x+y = 78, so y = 78-x.  We also want the product, xy, to be a maximum.  Let's substitute 78-x in place of y in the product:

 

xy = (78-x)x = -x² + 78x

 

This is a quadratic equation and its graph is a parabola.  Since the coefficient of the x² term is negative (-1), its an inverted parabola with the vertex at the top.  The vertex represents the maximum value of the product.  The vertex is always located at the point x = -b/2a, where b is the coefficient of the x term (78) and a is the coefficient of the x² term (-1).  So:

 

x = -78/2(-1) = 39

 

So y = 78-x = 39