A farmer wants to fence a small rectangular yard next to a barn. Fence for side parallel to the barn will cost 75 per foot and the fence for the other two sides will cost 30 per foot. The farmer has a total of 1750 dollars to spend on the project. Find the dimensions for the yard that will have the largest possible area.
The side parallel to the barn should be feet long and the other two sides should be feet long each.
Let's assume that the side parallel to the barn is x feet long, and the other two sides are y feet long each. The cost of the fence for the side parallel to the barn will be 75*x dollars, and the cost of the fence for the other two sides will be 30*y dollars each.
The total cost of the fence will be the sum of the costs of the two types of fence, which should be less than or equal to $1750:
75*x + 30*(2*y) <= 1750
Simplifying the above equation:
75*x + 60*y <= 1750
75*x + 60*y - 1750 <= 0
The area of the rectangular yard is given by the formula A = x * y.
To find the dimensions that will yield the largest possible area, we need to maximize the area function A, given the constraint of the total cost of the fence.
One method to find the maximum area is to use the method of Lagrange multipliers. However, in this case, we can use a simpler approach.
Since we are looking for the largest possible area, we can assume that the equality in the constraint equation holds:
75*x + 60*y - 1750 = 0
Now we have a system of two equations:
75*x + 60*y = 1750 (equation 1)
75*x = 1750 - 60*y (equation 2)
Simplifying equation 2:
x = (1750 - 60*y)/75
Substituting this into equation 1:
(1750 - 60*y)/75 + 60*y = 1750
Simplifying this equation:
1750 - 60*y + 75*60*y = 1750*75
1750 + 4500*y = 131250
4500*y = 129500
y = 129500/4500
y = 28.78
Substituting this value of y back into equation 1:
75*x + 60*28.78 = 1750
75*x = 1750 - 60*28.78
75*x = 1750 - 1726.8
75*x = 23.2
x = 23.2/75
x = 0.309
Therefore, the side parallel to the barn should be approximately 0.309 feet long, and the other two sides should be approximately 28.78 feet long each.
To find the dimensions for the yard that will have the largest possible area, we need to maximize the area of the rectangle. Let's assume the length of the side parallel to the barn is x feet.
Given:
Cost of fence parallel to the barn = $75 per foot.
Cost of fences on the other two sides = $30 per foot.
Total budget for the project = $1750.
The cost of the fence parallel to the barn can be calculated as 75x, and the cost of the other two sides is 30(2x) since there are two sides of the same length.
Total cost of the fence = Cost of fence parallel to the barn + Cost of the other two sides
Total cost = 75x + 30(2x) = 75x + 60x = 135x
We need to find the maximum value of x, such that the total cost (135x) does not exceed the total budget of $1750. Therefore, we can write the inequality:
135x ≤ 1750
To find the maximum value of x, divide both sides of the inequality by 135:
x ≤ 1750 / 135
Calculating the value on the right side gives us approximately x ≤ 12.96.
Since x represents the length of the side parallel to the barn, it should be a whole number, so we can take the largest whole number less than or equal to 12.96, which is x = 12.
So, the length of the side parallel to the barn should be 12 feet.
Since the other two sides are equal in length, each of them is also 12 feet.
Therefore, the dimensions of the yard that will have the largest possible area are:
The side parallel to the barn should be 12 feet long,
and the other two sides should be 12 feet long each.
If the expensive side has length x, and the other is y, then the cost is
75x+30*2y = 1750
The area is a = xy
Since y = (1750-75x)/60 = (350-15x)/12
a = x(350-15x)/12 = 1/12 (350x-15x^2)
This is just a parabola with vertex at x = 35/3.
That is x, so now you can figure y.