suppose that you have 800ft of fencing. you are to construct a rectangular corral which is divided into two pieces. what are the dimensions that give the largest possible areas?
The way to divide a rectangle in two pieces using minimum fence materials is to build a third fence segment having the width dimension, W,in the middle somewhere. The rectangle length L is then given by
3W + 2L = 800, so
L = 400 - 3/2 W
Area = L*W = 400W -(3/2)W^2
dA/dW = 0 = 400 - 3W
W = 133 1/3 ft ; L = 200
gives maximum area
To find the dimensions that give the largest possible area for the rectangular corral, we can use the concept of optimization. We will start by breaking down the problem into smaller steps:
Step 1: Define the variables:
Let's denote the length of one piece of the corral as x, and let y represent the width of the other piece.
Step 2: Formulate the objective function:
The objective is to maximize the area of the corral. Since the area of a rectangle is given by A = length × width, the objective function in this case is A = xy.
Step 3: Set up the constraint equation:
The perimeter of the corral is given as 800ft, and it consists of four sides (two lengths and two widths), which should add up to 800ft:
2x + 2y = 800.
Step 4: Solve for y in terms of x:
Rearrange the constraint equation to express y in terms of x:
2y = 800 - 2x,
y = 400 - x.
Step 5: Substitute the value of y into the objective function:
Substitute the expression for y into the objective function to get a single variable equation for A:
A = x(400 - x).
Step 6: Maximize the objective function:
To maximize the area, we need to find the critical points of the objective function. Differentiate A with respect to x and set it equal to zero to find its critical points:
dA/dx = 400 - 2x = 0.
Solving for x, we get x = 200.
Step 7: Calculate the corresponding y value:
Substitute the value of x = 200 into the constraint equation to find y:
2(200) + 2y = 800,
400 + 2y = 800,
2y = 400,
y = 200.
Therefore, by solving the above equations, we find that the dimensions that give the largest possible area for the rectangular corral divided into two pieces are x = 200ft and y = 200ft, or in other words, a square shape with sides of 200ft.