A rectangular lot adjacent to a highway is to be enclosed by a fence. If the fencing costs $2.50 per foot along the highway and $1.50 per foot on the other sides, find the dimensions of the largest lot that can be fenced off for $720.
Let the highway side have length x, and the rest will have total length x+2y.
We know that the cost is such that
2.50x + 1.50(x+2y) = 720
4x + 3y = 720
so, y = 240 - 4/3 x
The area a = xy = x(240 - 4/3 x) = 240x - 4/3 x^2
da/dx = 240 - 8/3 x
da/dx=0 when x = 90
So, the maximum area is 90 * 120 = 10800 ft^2
Well, fencing is all about having a good "perimeter" defense! So, let's break it down. We know that the cost along the highway is $2.50 per foot, and the cost on the other sides is $1.50 per foot. We want to maximize the area of the lot, while staying within a budget of $720.
Let's assume the length of the lot along the highway is "x" feet. Since we are enclosing a rectangular lot, the length opposite to the highway will also be "x" feet. Now, the remaining two sides will be y feet each.
So, the cost of fencing along the highway will be 2.5x dollars, and the cost of fencing the remaining two sides will be 2(1.5y) dollars.
According to the problem, the total cost is $720, so we can set up the equation:
2.5x + 2(1.5y) = 720
Now, let's isolate one variable and solve for it. Let's solve for y:
2(1.5y) = 720 - 2.5x
3y = 720 - 2.5x
y = (720 - 2.5x) / 3
Now, let's find the area of the lot. The area is given by length x width, which is x * y. Substituting the value of y, we get:
Area = x * ((720 - 2.5x) / 3)
To maximize the area, we'll take its derivative with respect to x and set it equal to zero:
d(Area) / dx = (720 - 2.5x)(1/3) - 2.5/3 * x = 0
Simplifying this equation, we'll get:
720 - 2.5x - (5/3)x = 0
(3/3)*720 = (2.5 + 5/3)x
2160 = (7.5 + 5/3)x
2160 = (22.5/3)x
Now, solving for x, we find:
x = (2160 * 3) / 22.5
x ≈ 288 feet
Substituting this value back into the equation for y, we get:
y = (720 - 2.5(288)) / 3
y ≈ 82 feet
Therefore, the dimensions of the largest lot that can be fenced off for $720 are approximately 288 feet by 82 feet.
Let's assume the dimensions of the rectangular lot are length (L) and width (W).
Since the fencing cost is $2.50 per foot along the highway side, and $1.50 per foot on the other sides, we can write the equation for the total cost of the fence as follows:
Total fencing cost = 2.50 * (L) + 1.50 * (W + L + W)
Since the given budget is $720, we can set up the equation:
2.50L + 1.50(W + L + W) = 720
Simplifying this equation, we get:
2.50L + 1.50W + 1.50L + 1.50W = 720
4L + 3W = 720
Now, to find the dimensions of the largest lot that can be fenced off, we need to maximize the area of the lot (L * W).
Since L and W are positive values, we can use the equation of the total fencing cost to solve for one variable in terms of the other and substitute it into the area equation.
Rearranging the total fencing cost equation, we get:
4L = 720 - 3W
L = (720 - 3W) / 4
Substituting this into the area equation, we get:
Area = L * W = [(720 - 3W) / 4] * W
To maximize the area, we can take the derivative of the area equation with respect to W, set it equal to 0, and solve for W.
d(Area)/dW = [(720 - 3W) / 4] - (3W / 4) = 0
Now, let's solve this equation:
720 - 3W - 3W = 0
720 - 6W = 0
6W = 720
W = 720 / 6
W = 120
Substituting this value of W back into the equation for L, we get:
L = (720 - 3 * 120) / 4
L = (720 - 360) / 4
L = 360 / 4
L = 90
Therefore, the dimensions of the largest lot that can be fenced off for $720 are a length of 90 feet and a width of 120 feet.
To find the dimensions of the largest lot that can be fenced off for $720, we need to set up an equation using the cost per foot of each type of fencing.
Let's assume that the length of the lot parallel to the highway is x feet, and the width of the lot perpendicular to the highway is y feet.
The cost of fencing along the sides parallel to the highway would be $2.50 * x per foot. Since there are two sides of length x, the total cost for these sides would be $2.50 * x * 2.
The cost of fencing along the other two sides (perpendicular to the highway) would be $1.50 * y per foot. Again, since there are two sides of length y, the total cost for these sides would be $1.50 * y * 2.
Adding up the costs of all four sides should be equal to $720, so we can set up the equation:
$2.50 * x * 2 + $1.50 * y * 2 = $720.
Simplifying the equation:
5x + 3y = 720.
To find the dimensions that maximize the area of the lot, we can use calculus or graphing techniques. However, in this case, let's solve it using a graphical method.
Let's solve the equation for y in terms of x:
3y = 720 - 5x,
y = (720 - 5x) / 3.
Now, let's plot the graph of this equation.
Once we have the graph, we can visually identify the dimensions (x, y) that maximize the area. The area of the lot can be found by multiplying x and y:
Area = x * y.
Finding the maximum point on the graph will give us the dimensions of the largest lot that can be fenced off for $720.