a barn is 150 feet long and 75 feet wide. the owner has 240 ft of fencing and wants to construct two adjacent outdoor pens of identical size along the long side of the barn using the barn as one side of each pen. the owner wants the pens as large as possible. what are the dimensions of the pens that will maximize the area?
If the width of the pens is x, and the overall length is y, then
3x+y = 240
The area is
a = xy = x(240-3x) = 240x - 3x^2
This is just a parabola, with vertex at x = 40
So, the pens are each 40 by 60 ft.
As usual, maximum area is achieved when the available fence is divided equally among lengths and widths.
To maximize the area of the pens, we need to find the dimensions that will use up the entire 240 feet of fencing.
Let's assume the width of each pen is x feet. Since we have two pens, the total length of the fencing used for the width of the pens is 2x.
Since the barn is already one side of each pen, we don't need any extra fencing for the length of the pens.
The remaining fencing, which is equal to 240 feet - 2x, will be used for the two widths of the pens. Since each pen has one width using the barn, we only need one additional width for each pen.
Therefore, the equation for the remaining fencing is:
240 feet - 2x = 2(150 feet) - x
Simplifying the equation:
240 feet - 2x = 300 feet - x
240 feet + x = 300 feet
x = 60 feet
Now that we have the value of x, we can calculate the dimensions of each pen:
Width = x = 60 feet
Length = 150 feet
So, the dimensions of each pen that will maximize the area are 60 feet by 150 feet.
To find the dimensions of the pens that will maximize the area, we can follow these steps:
Step 1: Visualize the problem
Draw a diagram to represent the barn and the two adjacent pens along the long side of the barn.
Let's label the length of each pen as 'x'. Since the barn's length is 150 feet, the combined length of the two pens along the barn will be 2x.
Step 2: Determine the fencing required
Each pen will have four sides: two lengths and two widths. Since the barn will act as one side of each pen, we will need fencing for the other three sides of each pen.
The dimensions of each pen are as follows:
Length = x feet
Width = 75 feet
For each pen:
Fencing required for the widths = 2 * (Width) = 2 * 75 = 150 ft
Fencing required for the lengths = 2 * (Length) = 2 * x = 2x ft
Total fencing required for each pen = Fencing for widths + Fencing for lengths = 150 + 2x ft
Since there are two pens, the total fencing required will be twice the amount:
Total fencing required = 2 * (Total fencing for each pen) = 2 * (150 + 2x) = 300 + 4x ft
Step 3: Find the expression for the area
The area of each pen can be calculated by multiplying the length and width:
Area = Length * Width = x * 75 = 75x ft²
To maximize the area, we need to maximize the value of 'x'.
Step 4: Write the constraint equation
The total amount of fencing available is given as 240 ft, which should be equal to the total fencing required for the pens (300 + 4x).
240 ft = 300 + 4x
Step 5: Solve the equation for 'x'
Now, let's solve the equation for 'x' to find the maximum area.
240 = 300 + 4x
4x = 240 - 300
4x = -60
x = -60/4
x = -15
Since we cannot have negative dimensions, this solution doesn't make sense in the context of the problem. We will proceed to evaluate the endpoints of the possible range for 'x'.
When x = 0:
Area = 75 * 0 = 0 ft²
When x = (240 - 300) / 4 = -15:
Area = 75 * (-15) = -1125 ft²
Again, negative area doesn't make sense, so we disregard this solution.
Step 6: Evaluate the remaining endpoint
When x = (240 - 300) / 4 = -60/4 = 15:
Area = 75 * 15 = 1125 ft²
The maximum area that can be obtained by constructing two adjacent pens along the long side of the barn is 1125 square feet when each pen has dimensions of 15 feet by 75 feet.
Note: It's important to verify if these dimensions are reasonable. In practical scenarios, negative dimensions or dimensions larger than the available space should be reviewed again.