A rectangular dog pen is constructed using a barn wall as one side and 50 meters of fencing for the other three sides. What is the maximum area of the dog pen?
length of side parallel to barn === y
length of each of other two sides === x
so we have 2x + y = 50
y = 50-2x
area = xy = x(50-2x)
= -2x^2 + 50x
We need the vertex of this parabola
the x of the vertex = -50/-4 = 12.5
then y = 50-2(12.5) = 25
so max area = xy = 12.5(25) m^2 or 312.5 m^2
You could also find x by completing the square,
or by Calculus setting the derivative of -2x^2 + 50x equal to zero
To find the maximum area of the rectangular dog pen, we need to determine the dimensions of the rectangle that will give us the largest possible area.
Let's assume the barn wall is the length of the rectangle, and the other two sides are the width of the rectangle.
Let's say the length of the rectangle is L and the width is W.
Given that one side of the rectangle is the barn wall, we have L = 50 meters.
We also know that the perimeter of the rectangle is 50 meters.
Perimeter of a rectangle = 2L + 2W
Substituting the values, we get:
50 = 2L + 2W
Since L = 50, we can substitute this value into the equation:
50 = 2(50) + 2W
50 = 100 + 2W
2W = 50 - 100
2W = -50
W = -25
Since width cannot be negative, it is not a valid solution.
To find the maximum area, we need to consider a case where the width is as large as possible.
Let's set W = 25.
Now, we can calculate the area of the rectangle using the formula:
Area of a rectangle = Length * Width
Area = L * W
Area = 50 * 25
Area = 1250 square meters
Therefore, the maximum area of the dog pen is 1250 square meters.
To find the maximum area of the dog pen, we need to determine the dimensions that would maximize the area. Let's say the length of the pen parallel to the barn wall is x meters and the width perpendicular to the barn wall is y meters.
Since one side of the pen is the barn wall, we only need to consider the other three sides for the fencing. Thus, the perimeter of the pen is given as:
Perimeter = 2x + y = 50 meters
To express y in terms of x, we can solve the above equation for y:
y = 50 - 2x
Now, the area of the pen (A) is given by:
Area = length × width = x × y
Substituting the value of y that we obtained:
Area = x(50 - 2x)
To find the maximum area, we need to find the value of x that maximizes the above expression. This can be done by finding the vertex of the parabola given by the equation. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In this case, a = -2, b = 50, and c = 0. Plugging in these values, we have:
x = -50 / (2×(-2))
x = -50 / (-4)
x = 12.5
Therefore, x = 12.5 maximizes the area of the pen.
To find the corresponding value of y, we can substitute the value of x back into the equation we derived earlier:
y = 50 - 2x
y = 50 - 2(12.5)
y = 50 - 25
y = 25
Thus, the dimensions that maximize the area of the dog pen are x = 12.5 meters and y = 25 meters. Finally, we can calculate the maximum area by substituting these values into the formula for area:
Maximum Area = x × y
Maximum Area = 12.5 × 25
Maximum Area = 312.5 square meters
Therefore, the maximum area of the dog pen is 312.5 square meters.