you have 20 feet of fencing material to make a pen for your dog The design of the pen is given as follows
To make the most use of the 20 feet of fencing material, we can create a rectangular pen for your dog. Since the design is not given in the question, we will assume it is a standard rectangular shape.
In a rectangular pen, opposite sides are equal in length. Let's represent the length of the pen as L and the width as W.
We need to find the dimensions of the pen that maximize the enclosed area (the area your dog will have). The perimeter of a rectangular pen is given by the equation:
Perimeter = 2L + 2W
Given that we have 20 feet of fencing material, we can write:
2L + 2W = 20
Dividing the equation by 2 gives us:
L + W = 10
Now, to maximize the enclosed area, we need to find the dimensions that satisfy this equation and lead to the largest possible area.
One way to approach this is by rewriting the equation as:
L = 10 - W
Now, substitute the value of L from this equation into the formula for the area of a rectangle:
Area = L * W
Replacing L with 10 - W, we get:
Area = (10 - W) * W
Expanding the equation:
Area = 10W - W^2
Now, we need to find the value of W that maximizes the area. This can be achieved by finding the vertex of the quadratic equation.
The area is maximized when the coefficient of the squared term is negative. In this case, since the coefficient of W^2 is -1, we can conclude that the area will be maximized when:
W = -b / 2a
Plugging in the values a = -1 and b = 10 into the formula, we get:
W = -10 / 2(-1)
W = -10 / -2
W = 5
Therefore, the width of the rectangular pen that maximizes the area is 5 feet. To find the length, we substitute this value back into the equation L = 10 - W:
L = 10 - 5
L = 5
So, the length of the rectangular pen is also 5 feet.
In conclusion, to make the most use of the 20 feet of fencing material, you should construct a rectangular pen with dimensions 5 feet by 5 feet.