Old McDonald has 1000 feet of fencing, and wants to build a pen along the side of his barn that will have two separate areas- one for his pigs, and one for his cows. if only one of the fences is to run parallel to the barn, find the dimensions of the pen that would maximize the area.
assuming a rectangular fenced area, let
parallel length = x
area = x(1000-x)/3 = 1/3 (1000x-x^2)
Just figure where the vertex of the parabola is, and that will give you the maximum area.
where did you get the 1/3 from
Assume the length of the fence opposite the barn side is: x.
The remaining (1000-x) feet of fence work is divided into three by the two other sides and the partition. Each being: (1000-x)/3.
The area of the pen is: y = x(1000-x)/3
The area is maximum when the derivative is zero. The derivative is:
y' = (1000-2x)/3
Solve for x when y' = 0.
since this was pre-calc, I assumed that calculus was not available. But the parabola's vertex is from back in Algebra I!
To find the dimensions of the pen that would maximize the area, we can break down the problem into smaller steps:
Step 1: Define the variables.
Let's assume the length of the pen parallel to the barn as 'x' feet. This means the side fences will have a length of 'x' each, and the remaining fencing will be used as the fencing between the two separate areas. So, the length of this fence will be 1000 - 3x feet.
Step 2: Formulate the equation for the area.
The area of the pen is determined by the product of its length and width. Since one side of the pen is defined as parallel to the barn, the width will be 'y' feet. Thus, the area can be expressed as: A = x * y.
Step 3: Express the width in terms of 'x'.
To express the width 'y' in terms of 'x', we need to consider that the fencing between the two separate areas will divide the width in half. So, the width will be (1000 - 3x) / 2.
Step 4: Replace the width in the area equation.
Substitute the expression for 'y' in terms of 'x' into the area equation: A = x * ((1000 - 3x) / 2).
Step 5: Simplify the equation.
Simplify the equation by multiplying both sides by 2 to eliminate the fraction: 2A = x * (1000 - 3x).
Step 6: Maximize the area equation.
To find the dimensions that maximize the area, we need to find the maximum value for the equation. In this case, we want to find the maximum area (A), so we need to find the maximum value for 2A.
Step 7: Use calculus or graphing to find the maximum.
Differentiating the equation 2A = x * (1000 - 3x) with respect to 'x' gives: d(2A)/dx = 1000 - 6x. Setting this derivative equal to zero, we find 1000 - 6x = 0. Solving for 'x', we get x = 1000 / 6 = 166.67 feet.
Step 8: Substitute 'x' into one of the expressions to find 'y'.
Substitute the value of 'x' into either expression for 'y' we derived earlier. Using (1000 - 3x) / 2, we get y = (1000 - 3 * 166.67) / 2 = 833.33 / 2 = 416.67 feet.
Therefore, to maximize the area of the pen, the dimensions should be approximately 166.67 feet parallel to the barn and 416.67 feet perpendicular to the barn.