Algebra question Vertex formula y = a(x – h)^2 + k?

Question

John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.

Show clearly the algebraic steps which prove your dimensions are the maximum area which can be obtained. Use the vertex formula to find the maximum area.

vertex formula
y = a(x – h)^2 + k


note----(2 is to the second power )

Answer 1

2l + 2w = 300 which is l + w = 150

l = 150-w

A = l*w

A = (150-w)*w

A = 150w - w^2

-A = w^2 - 150w (multiply by -1)

w^2 - 150w + 75^2 = -A + 75^2 (complete the square)

(w-75)^2 = -A + 75^2 (show the square)

-(w-75)^2 = A - 75^2 (multiply by -1)

-(w-75)^2 + 75^2 = A (solve for A)

So if your vertex is (h,k) for A = lw

Vertex = (75,75^2) implying both l and w are 75