a farmer has 120 m of fencing to make two identical rectangular enclosures using an existing wall as one side of each enclosure. The dimensions of each closure are x metres and y metres as shown. Obtain and expression in terms of x only for the total area of two enclosures, and calculate the maximum value of the area. I"m really stuck please help!
assuming the x side is parallel to the wall, then
2x+4y = 120
The area is
2xy = 2x(120-2x)/4 = 60x - x^2
The maximum area is thus where x=30
As usual with such problems, the maximum area is achieved when the fencing is divided equally among lengths and widths.
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To solve this problem, let's go step by step.
Step 1: Understand the problem
We have a farmer who wants to create two identical rectangular enclosures using a wall as one side of each enclosure. The total amount of fencing available is 120 meters. We need to find an expression for the total area of the enclosures in terms of x and y, find the maximum value of the area, and express it in terms of x only.
Step 2: Determine the dimensions
Let's assume that the width of each enclosure is x meters and the length is y meters. Notice that we have two enclosures, so we need to consider the area of both enclosures combined.
Step 3: Understand the constraints
We know that the total amount of fencing available is 120 meters. Since we have two enclosures, we would need to consider the fencing required for the sides and the fence between the two enclosures as well.
For each enclosure, we have two equal sides with length y and one side with length x. Additionally, we have two fences of length x between the two enclosures. This gives us the total length of fencing as:
2y + x + 2x = 120
2y + 3x = 120
Step 4: Express the area in terms of x and y
The area of a rectangle is given by the formula: Area = Length * Width.
For each enclosure, the area would be:
Area_enclosure = y * x
Since we have two identical enclosures, the total area is:
Total_Area = 2 * Area_enclosure = 2 * y * x
Step 5: Express the total area in terms of x only
Using the equation from step 3, 2y + 3x = 120, we can solve for y:
2y = 120 - 3x
y = (120 - 3x)/2
Now, substitute this value of y in the expression for Total_Area:
Total_Area = 2 * (120 - 3x)/2 * x
Total_Area = (120 - 3x) * x
Total_Area = 120x - 3x^2
Step 6: Find the maximum value of the area
To find the maximum value of the area, we need to find the maximum value of the quadratic equation Total_Area = 120x - 3x^2.
This can be done by finding the vertex of the quadratic equation, which is given by the formula:
x = -b/2a
In this case, a = -3 and b = 120, so the x-coordinate of the vertex is:
x = -120/(2*(-3))
x = -120/(-6)
x = 20
Therefore, the maximum value of the area is obtained when x = 20.
To find the corresponding y-value, substitute x = 20 in the equation for y:
y = (120 - 3(20))/2
y = (120 - 60)/2
y = 60/2
y = 30
So, when x = 20, the maximum area of two enclosures is:
Total_Area = 120(20) - 3(20^2)
Total_Area = 2400 - 3(400)
Total_Area = 2400 - 1200
Total_Area = 1200 square meters.
Therefore, the maximum area of the two enclosures is 1200 square meters, which occurs when the width of each enclosure is 20 meters and the length is 30 meters.