Finding The Maximum Using The Vertex Worksheet With Answer Key
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5Section 3.3B Word Problems: Maximizing (Finding the Maximum using the Vertex)
Example1:
At a concert, organizers are roping off a rectangular area for sound equipment. There is 160m
of fencing available to create the perimeter. What dimensions will give the maximum area, and
what is the maximum area?
Steps:
1) Write an equation for the perimeter, and write an equation for the area for the rectangle.
2) Use the two equations to create a quadratic function in standard form.
3) Change the quadratic function into vertex-graphing form.
4) Identify the maximum area, and then the dimensions for the maximum area.
y
1) If we let x be the width and y be the length, A is the area.
Sound
x
x
Equipment
Perimeter = 2x + 2y with x and y being the width and length
So 160 = 2x + 2y (Solve for y to create a function.)
y
160 – 2x = 2y
80 – x = y
y = 80 – x
Area = xy
A = xy
2) Now combine the 2 equation from part 1) to find an equation to maximize the area.
y = 80 – x
A = xy
A = x(80 – x)
2
A = 80x – x
2
A = -x
+ 80x
b
3) Now complete the square to change into vertex-graphing form OR use
=
−
formula.
p
2a
Since all we need is the vertex, the simpler and quicker way is to use the formula to find the
2
vertex. A = -x
+ 80x
a = -1, b = 80
b
80
=
−
=
−
=
So the vertex is (40, ??)
(x, A)
p
40
−
2a
( 2
) 1
Substitute 40 in for x and we get:
2
A = -(40)
+ 80(40) = -1600 + 3200 = 1600
Our vertex is (40, 1600)
2
4) Using the vertex, our maximum area is 1600m
and one dimension of the box is 40m.
Since A = xy 1600=40y so y = 40 The box is 40m by 40m.
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