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Answer :
1. The largest possible area he can enclose is 18000 = x + y. So, the correct answer is A. 18000 = x + y
2.the maximum area is A(x) = 4500x² . So, the correct answer is A.
A(x) = 4500x²
Let's break down the problem step by step.
Setting up the equation:The farmer has 18,000 linear feet of fencing to construct a pen. Let's assume the length of the pen is x feet and the width is y feet. The perimeter of the pen (which is the total length of fencing used) can be represented by the equation: Perimeter = 2x + 2y
Since the total length of fencing is 18,000 feet, we can write the equation as: 2x + 2y = 18000
Now, we can simplify this equation by dividing both sides by 2: x + y = 9000
So, the correct equation is 18000 = x + y, which corresponds to option A.
2.Calculating the maximum area:The area of a rectangular pen can be calculated using the formula: Area = Length × Width
Area = x × y
We want to maximize the area while keeping the perimeter (fencing length) constant at 18,000 feet. From the first equation, we know that x + y = 9000, so we can solve for y: y = 9000 - x
Substituting this value of y into the area formula: Area = x × (9000 - x)
Now we have an equation for the area in terms of x. To find the maximum area, we need to find the value of x that maximizes this equation. This is a quadratic equation, and the maximum occurs at the vertex of the parabola.
The formula for the x-coordinate of the vertex of a parabola of the form ax² + bx + c is given by: x_vertex = -b / (2a)
In our case, the equation for the area is -x² + 9000x. Comparing this to ax² + bx + c, we have a = -1, b = 9000, and c = 0. Plugging these values into the formula: x_vertex = -9000 / (2 * -1)
x_vertex = 4500
So, the maximum area occurs when x = 4500 feet.
Now, we can calculate the corresponding y value using the equation y = 9000 - x: y = 9000 - 4500
y = 4500
Therefore, the maximum area is:Area = x × y
Area = 4500 × 4500
Area = 20,250,000 square feet
So, the correct equation for the maximum area is A(x) = 4500x² which corresponds to option A.
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Correct Question is:
31 wall river 2 set up the equation : If a farmer has 18,000 linear it of fencing with which to construct his pens,
1. what is the largest possible area he can enclose ?
A. 18000 = x + y
B. 18000 = 4x +27
C. 18000=2x+4y
D. 18000 = 3x+2y
E. 18000=2x+3y 3
2. calculate the max area, area=
A. A(x) = 4500x-27
B. A(x) = 4 500x - 1 3 x 2
C.A(x) = 60000 - 2x²
D. A(x) = 6000x - 2 x ²
E. A(x) = 9000x-2 ta २ 3 2 2 .ft 2
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