Quadratic function word problem

Question

a)Find the area of the rectangle of maximum area with perimeter 12 centimeters. 

 

b) Find two real numbers whose difference is 10 and whose product has the least possible value.

 

For a I got Max area = 4

for b I got x=5, y=-5

 

If I got them right great, but I didn't really understand what I was doing. So if I got them wrong, please explain

Answer 1

a).  Let x = length and y = width

 

       Then, 2x + 2y = 12    So, y = 6 - x

 

       A = area = xy = x(6 - x) = -x² + 6x, 0 < x < 6

 

       The graph of the area function is a parabola opening downward.  It has its maximum value at the vertex (3,9).

 

       Maximum area = 9 cm²

 

b).  If the two numbers are x and y, then x - y = 10, so y = x - 10.

 

      Minimize P = xy = x(x - 10) = x² - 10x

 

      The graph of P is a parabola opening upward with vertex at (5,-25).

 

      The numbers are 5 and -5.  The minimum product is -25.