A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $7 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $15,400?
perimeter = 2 x + 2 y
cost = c = 14 x + 4 y = 15400
or
7 x + 2 y = 7700
y = -3.5 x + 3850
A = x y = x(-3.5 x + 3850)
= -3.5 x^2 + 3850 x
dA/dx = -7x +3850
zero when
x = 3850/7
etc
Ah, the art of fencing! Let's calculate the dimensions of this rectangular plot with the greatest area while keeping the cost in check.
Let's say the heavy-duty fencing, which costs $7 per foot, is used for the longer sides of the rectangular plot. The standard fencing, which costs $2 per foot, will be used for the remaining sides.
Let's denote the length and width of the rectangular plot as L and W, respectively. We know that the cost of fencing is given by:
2L + 2W = $15,400
Furthermore, the area of the rectangular plot, A, is given by:
A = L * W
Now, let's solve these equations with a touch of mathematical magic and a pinch of pizzazz!
Since we decided to use heavy-duty fencing for the longer sides, let's assume that L is greater than W. Therefore, we can rewrite the cost equation as:
2L + W + W = $15,400
2L + 2W = $15,400
L + W = $7,700
Now, let's express L in terms of W by subtracting W from both sides:
L = $7,700 - W
Substituting L in the area equation, we get:
A = ($7,700 - W) * W
To find the maximum area, we'll need a bit of calculus. Let's take the derivative of A with respect to W and set it equal to zero:
dA/dW = 0
($7,700 - W) - W = 0
$7,700 - 2W = 0
2W = $7,700
W = $7,700/2
W = $3,850
Now that we've found W, we can substitute it back into the cost equation to find L:
L + $3,850 = $7,700
L = $7,700 - $3,850
L = $3,850
So, the dimensions of the rectangular plot with the greatest area that can be fenced in at a cost of $15,400 are approximately 3850 feet by 3850 feet.
Keep in mind, my calculations are as sharp as a rubber knife, so please double-check the math just to be safe!
Let's assume the length of the rectangular plot is L and the width is W.
The cost of heavy-duty fencing per foot is $7, and the cost of standard fencing per foot is $2.
We are given that the cost of fencing is $15,400, so the total cost of the heavy-duty fencing is $15,400 - 2 * $2 * W - 2 * $2 * L.
We want to maximize the area of the rectangular plot, which is given by the formula A = L * W.
To solve this problem, we need to express the area A in terms of a single variable.
Since we're given the total cost, we can rewrite the equation for the total cost as follows:
$15,400 = 2 * $7 * W + 2 * $2 * L
Simplifying the equation, we get:
$15,400 = $14 * W + $4 * L
Simplifying further, we get:
$15,400 = 14W + 4L
Now we need to express L or W in terms of a single variable. Let's solve for L:
4L = $15,400 - 14W
L = ($15,400 - 14W)/4
Now we can express the area in terms of a single variable, W:
A = W * (($15,400 - 14W)/4)
To find the dimensions of the rectangular plot that maximize the area, we can differentiate the area equation with respect to W and set it equal to zero:
dA/dW = (($15,400 - 14W)/4) + (W / -2) = 0
Simplifying the equation, we get:
($15,400 - 14W) - 2W = 0
15,400 - 16W = 0
16W = 15,400
W = 15,400/16
W = 962.5
Now we can substitute the value of W back into the equation for L to find L:
L = ($15,400 - 14 * 962.5)/4
L = $15,400 - 13,475/4
L = $15,400 - 3,368.75
L = $12,031.25
Therefore, the dimensions of the rectangular plot that maximize the area are 962.5 ft by 12,031.25 ft.
To find the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $15,400, we can start by setting up some equations.
Let's assume the length of the rectangular plot is "x" feet, and the width is "y" feet.
We know that two opposite sides will use heavy-duty fencing, which costs $7 a foot, and the other two sides will use standard fencing, which costs $2 a foot.
The cost of the heavy-duty fencing is given by: 2x * 7 = 14x dollars.
The cost of the standard fencing is given by: 2y * 2 = 4y dollars.
The total cost of the fencing is $15,400. So, we can write the equation: 14x + 4y = 15,400.
Now, let's consider the area of the rectangular plot. The area is given by: A = x * y.
To find the dimensions of the rectangular plot of greatest area, we need to maximize the area A.
To maximize the area, we can use the concept of optimization. We have an equation (14x + 4y = 15,400) that represents a constraint on the cost of the fencing. We can solve this equation for one variable (for example, y) and substitute it into the equation for the area A. Then, we can find the derivative of A with respect to x, set it equal to zero, and solve for x.
Let's proceed with the calculations to find the dimensions.