A Farmer Has 100 M Of Fencing To Enclose A Rectangular Pen.Which Quadratic Equation Gives The Area (A) Of The Pen, Given Its Width (w)?A. A ( W ) = W 2 − 50 W A(w) = W^2 - 50w A ( W ) = W 2 − 50 W B. A ( W ) = W 2 − 100 W A(w) = W^2 - 100w A ( W ) = W 2 − 100 W C. A ( W ) = 50 W − W 2 A(w) = 50w - W^2 A ( W ) = 50 W − W 2 D. $A(w) = 100w
Introduction
In this article, we will explore a classic problem in mathematics that involves a farmer with 100 meters of fencing to enclose a rectangular pen. The problem requires us to find the area of the pen, given its width. We will use algebraic techniques to derive the quadratic equation that represents the area of the pen.
The Problem
A farmer has 100 meters of fencing to enclose a rectangular pen. The farmer wants to know the area of the pen, given its width. Let's denote the width of the pen as w. We need to find the area of the pen, denoted as A, in terms of w.
The Solution
To solve this problem, we need to use the fact that the perimeter of the pen is equal to the total length of the fencing. The perimeter of a rectangle is given by the formula P = 2l + 2w, where l is the length and w is the width. Since the farmer has 100 meters of fencing, we can set up the equation:
2l + 2w = 100
We can simplify this equation by dividing both sides by 2:
l + w = 50
Now, we need to find the area of the pen, which is given by the formula A = lw. We can substitute the expression for l from the previous equation into the area formula:
A = (50 - w)w
Expanding the right-hand side of this equation, we get:
A = 50w - w^2
The Quadratic Equation
The equation we derived in the previous section is a quadratic equation in terms of w. It represents the area of the pen, given its width. We can write this equation in the standard form of a quadratic equation:
A(w) = 50w - w^2
This equation is a quadratic function of w, and it represents the area of the pen as a function of its width.
Conclusion
In this article, we solved a classic problem in mathematics that involved a farmer with 100 meters of fencing to enclose a rectangular pen. We used algebraic techniques to derive the quadratic equation that represents the area of the pen, given its width. The equation we derived is A(w) = 50w - w^2, which is a quadratic function of w.
Discussion
Which of the following quadratic equations gives the area (A) of the pen, given its width (w)?
A. B. C. D.
The correct answer is C. . This equation represents the area of the pen as a function of its width, and it is the correct solution to the problem.
Additional Information
- The equation A(w) = 50w - w^2 is a quadratic function of w, and it represents the area of the pen as a function of its width.
- The equation can be rewritten in the standard form of a quadratic equation: A(w) = -w^2 + 50w.
- The equation has a vertex at the point (25, 625), which represents the maximum area of the pen.
- The equation has two solutions, w = 0 and w = 50, which represent the minimum and maximum widths of the pen, respectively.
Final Answer
Introduction
In our previous article, we explored a classic problem in mathematics that involved a farmer with 100 meters of fencing to enclose a rectangular pen. We used algebraic techniques to derive the quadratic equation that represents the area of the pen, given its width. In this article, we will answer some frequently asked questions related to this problem.
Q&A
Q: What is the perimeter of the pen?
A: The perimeter of the pen is equal to the total length of the fencing, which is 100 meters.
Q: How do we find the area of the pen?
A: We can find the area of the pen by using the formula A = lw, where l is the length and w is the width. We can substitute the expression for l from the equation l + w = 50 into the area formula.
Q: What is the quadratic equation that represents the area of the pen?
A: The quadratic equation that represents the area of the pen is A(w) = 50w - w^2.
Q: What is the vertex of the quadratic equation?
A: The vertex of the quadratic equation is at the point (25, 625), which represents the maximum area of the pen.
Q: What are the solutions to the quadratic equation?
A: The solutions to the quadratic equation are w = 0 and w = 50, which represent the minimum and maximum widths of the pen, respectively.
Q: How do we determine the maximum area of the pen?
A: We can determine the maximum area of the pen by finding the vertex of the quadratic equation. The vertex represents the maximum area of the pen.
Q: What is the minimum width of the pen?
A: The minimum width of the pen is 0 meters.
Q: What is the maximum width of the pen?
A: The maximum width of the pen is 50 meters.
Q: How do we find the length of the pen?
A: We can find the length of the pen by using the equation l + w = 50. We can substitute the value of w into this equation to find the length of the pen.
Q: What is the relationship between the width and the length of the pen?
A: The width and the length of the pen are related by the equation l + w = 50. This equation represents the fact that the perimeter of the pen is equal to the total length of the fencing.
Conclusion
In this article, we answered some frequently asked questions related to the problem of a farmer with 100 meters of fencing to enclose a rectangular pen. We used algebraic techniques to derive the quadratic equation that represents the area of the pen, given its width. We also discussed the vertex and solutions of the quadratic equation, as well as the relationship between the width and the length of the pen.
Additional Information
- The equation A(w) = 50w - w^2 is a quadratic function of w, and it represents the area of the pen as a function of its width.
- The equation can be rewritten in the standard form of a quadratic equation: A(w) = -w^2 + 50w.
- The equation has a vertex at the point (25, 625), which represents the maximum area of the pen.
- The equation has two solutions, w = 0 and w = 50, which represent the minimum and maximum widths of the pen, respectively.
Final Answer
The final answer is C. .