Answer:
- see the attachments for the sketch and the graph
- A = x(120 -2x)
- x = 30 maximizes the area
- dimensions: 30 m by 60 m
- domain of x: [0, 60]
Step-by-step explanation:
You want an expression for area, maximum area, and dimensions of a rectangular pen fenced on three sides with a total of 120 m of fencing.
a) Sketch
The first attachment shows a sketch of the pen with x defined as the dimension the pen extends from the barn wall. The total length of fence is 120 m, so the side parallel to the barn will be (120 -2x) m in length.
b) Area
The area is the product of the length and width of the rectangle:
A = (x)(120 -2x)
c) Graph
The second attachment shows a graph of the area as a function of x.
d) Greatest area
The value of x that gives the greatest area is the x-coordinate of the vertex of the parabola. It is halfway between the zeros at x=0 and x=60. The maximum area will be had when x=30.
e) Dimensions
The dimensions of the pen are ...
x = 30
120 -2x = 120 -2(30) = 60
The pen is 30 m by 60 m, with the 60 m dimension parallel to the barn wall.
f) Domain
The area function only makes sense for values of x between 0 and 60 meters.
The domain of x is 0 ≤ x ≤ 60.
