A farmer has 54m of fencing with which to build two animal pens with a common side. one pen is rectangular; the other is square. if the area of the pens is to be maximized, what are their dimensions?
let the side of the square be x
let the sides of the rectangle be x by y
so 5x + 2y = 54
y = (54-5x)/2
area = x^2 + xy
= x^2 + x(54-5x)/2
= x^2 + 27x - (5/2)x^2
d(area)/dx = 2x + 27 - 5x
= 0 for a max of area
3x = 27
x = 9
y = (54 - 45)/2 = 4.5
the square is 9 by 9, and the rectangle is 9 by 4.5
bur
To maximize the area of the pens, we need to find the dimensions that will maximize the area of each pen.
Let's assume the rectangular pen has length L, width W, and the square pen has side length S.
We know that the rectangular pen has two equal sides (common side) and two different sides. Therefore, the fencing required for the rectangular pen can be calculated as:
2L + W = 54m -----(Equation 1)
The square pen has 4 equal sides. Therefore, the fencing required for the square pen can be calculated as:
4S = 54m -----(Equation 2)
We can solve these equations simultaneously to find the dimensions of each pen.
From Equation 2, we can solve for S:
4S = 54m
S = 54m / 4
S = 13.5m
Now, substitute the value of S into Equation 1:
2L + W = 54m
2L + 13.5m = 54m
2L = 54m - 13.5m
2L = 40.5m
L = 40.5m / 2
L = 20.25m
Now, substitute the values of L and S into Equation 1 to find W:
2L + W = 54m
2(20.25m) + W = 54m
40.5m + W = 54m
W = 54m - 40.5m
W = 13.5m
Therefore, the dimensions of the rectangular pen with the maximum area are:
Length = 20.25m
Width = 13.5m
And the dimensions of the square pen with the maximum area are:
Side length = 13.5m
To find the dimensions that maximize the area of the pens, we'll solve this problem step by step:
Step 1: Understand the problem
We have 54 meters of fencing available to build two animal pens - one rectangular and one square. The rectangular pen and square pen share a common side. We need to find the dimensions (length, width, and side length) that maximize the total area of both pens.
Step 2: Identify variables
Let's assign variables to the dimensions:
- Length of the rectangular pen: L (in meters)
- Width of the rectangular pen: W (in meters)
- Side length of the square pen: S (in meters)
Step 3: Formulate equations
We know the total length of fencing available is 54 meters, which can be expressed as:
2L + 3W + S = 54
(This accounts for the two lengths of the rectangular pen, three widths of the rectangular pen, and one side of the square pen.)
Step 4: Simplify equations
From the equation we formed, we can express S in terms of L and W:
S = 54 - 2L - 3W
Step 5: Determine the area
The area of the rectangular pen can be calculated as:
Area_R = L * W
The area of the square pen can be calculated as:
Area_S = S^2 = (54 - 2L - 3W)^2
The total area, Area_Total, is the sum of the areas of both pens:
Area_Total = Area_R + Area_S = L * W + (54 - 2L - 3W)^2
Step 6: Maximize the area
To maximize the total area, we differentiate the Area_Total equation with respect to L and W, then set both derivatives equal to zero. Solving these equations will give us the optimal values for L and W.
Step 7: Solve for optimal dimensions
By differentiating the Area_Total equation and solving for L and W simultaneously, we can find the dimensions that maximize the total area.
Note: The process of solving the differentiation equations can be complex and involve multiple steps. It would be ideal to use a symbolic math software or calculator to obtain the exact dimensions.