A rectangular pasture is subdivided into two equal pens. Using the barn as the LEFT side and 132m of fencing for the rest, find the value of x that gives the maximum area, and A(x).
How do i do this? What are the steps?
I just asked this same exact question yesterday, maybe we're in the same class! lol
THIS IS WHAT I WROTE + REPLIES:
"A rectangular pasture is subdivided into two equal pens. Using the barn as one side and 132 m of fencing for the rest, find the value of x that gives the maximum area, and A(x)."
It gives no diagram whatsoever, so I have no idea if all the sides are the same length or not, etc etc... Can someone show me how to work out this problem please? Any help is GREATLY appreciated!! :D
Precalculus - MathMate, Tuesday, September 22, 2009 at 12:33am
The fence will look like a letter E. The open end of the letter E is the face of the barn.
It does not matter which length x stands for, as long as the total length of the fence is 132 m.
Let x be one of the three equal sides, and the length of the barn fenced in is 132-3x.
Total area
A(x) = x(132-3x)
A'(x) = 132-6x =0
Therefore x=132/6=22 m
The area is x(132-3x)=22(66)=1452 m²
check: A"(x) = -6 <0, therefore maximum.
shouldnt this: x(132-3x) equal 132x-3x^2??? And NOT 132-6x =0 ? im confused!
nevermind i understand now!
x(132-3x) is the same exact thing as 132x-3x^2. x(132-3x) represents the area of the two pens together, with the length being 132-3x and the width being x.
oh okay :)
To find the value of x that gives the maximum area of the rectangular pasture, you need to follow these steps:
Step 1: Visualize the problem
Start by visualizing the rectangular pasture divided into two equal pens. Let's call the width of one pen x and the length of each pen 2x since the entire pasture is divided equally.
Step 2: Identify the constraints
The problem mentions that the left side of the rectangular pasture is the barn, and 132m of fencing is available for the remaining sides. This information helps establish a constraint on the perimeter of the pasture, which will later be used to write an equation.
Step 3: Express the area as a function of a single variable
The area of a rectangle can be calculated by multiplying its length by its width. In this case, since we are considering the width as x and the length as 2x, the area can be expressed as A(x) = 2x * x = 2x^2.
Step 4: Write an equation using the constraint
Since 132 meters of fencing is available for the remaining three sides, the perimeter of the rectangular pasture can be expressed as:
2x + 2(2x) = 132
Simplifying the equation, we get:
2x + 4x = 132
6x = 132
Step 5: Solve the equation for x
Divide both sides of the equation by 6 to solve for x:
x = 22
Step 6: Calculate the maximum area
Substitute the value of x (22) into the expression for the area to find the maximum area:
A(x) = 2(22)^2
A(x) = 2 * 484
A(x) = 968
Therefore, when x equals 22, the rectangular pasture will have the maximum area, which is 968 square meters.