the complete question is
Farmer brown built a rectangular pen for his chickens using 12 meters of fence. • he used part of one side of his barn as one length of the rectangular pen. • he maximized the area using the 12 meters of fence. farmer johnson built a rectangular pen for her chickens using 16 meters of fence. • she used part of one side of her barn as one length of the rectangular pen. • the length of her pen was 2 meters more than the length of farmer brown's pen. • the width of her pen was 1 meter more than the width of Farmer Brown’s pen.
How much larger is Farmer Johnson’s rectangular pen than Farmer Brown’s?
Lety------> the length of the rectanglex-----> the width of the rectangle
step 1
find the dimensions of the Farmer brown
we know that
perimeter of the rectangle=2*[x+y]
Since the barn is used as one of the sides (let's say y) we can subtract y
we don't need fencing for this side
That makes the perimeter 2x + y
Since we have 12 meters of fencing
we set these equal:
2x + y = 12-------> y=(12-2x)------> equation 1
area of the rectangle=x*y
substitute the equation 1 in the area formula
Area=[(12-2x)]*x-----> Area=(-2x²)+12x
This is a quadratic equation
Since the leading coefficient is negative (-2) we know it opens downward
We are looking for the x-coordinate of the highest point called the vertex
using a graph tool
see the attached figure
the vertex is the point (3,18)
that means for x=3 (width of the rectangle)
the area is 18 m²
the dimensions of the Farmer brown are
6 m x 3 m
length=6 m
width=3 m
step 2
find the dimensions of the Farmer Johnson
perimeter=16 m
length=6+2=8 m
width=1+3=4 m
step 3
How much larger is Farmer Johnson’s rectangular pen than Farmer Brown’s?
Farmer Johnson’s rectangular pen
Area=8*4=32 m²
Farmer Brown’s rectangular pen
Area=6*3=18 m²
32 m²-18 m²=14 m²
the answer is 14 m²
