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 - 50w $}$B. { A(w) = W^2 - 100w $} C . \[ C. \[ C . \[

by ADMIN 214 views

A Farmer's Fencing Problem: Finding the Area of a Rectangular Pen

As a farmer, you have 100 meters of fencing to enclose a rectangular pen. The goal is to determine the area of the pen, given its width. This problem involves using quadratic equations to model the relationship between the width and the area of the pen. In this article, we will explore the quadratic equation that gives the area of the pen, given its width.

To start, let's consider the dimensions of the rectangular pen. We know that the perimeter of the pen is equal to the total length of the fencing, which is 100 meters. The perimeter of a rectangle is given by the formula:

P = 2l + 2w

where P is the perimeter, l is the length, and w is the width.

Since we have 100 meters of fencing, we can set up the equation:

100 = 2l + 2w

We can simplify this equation by dividing both sides by 2:

50 = l + w

This equation tells us that the sum of the length and width of the pen is equal to 50 meters.

The area of a rectangle is given by the formula:

A = lw

where A is the area, l is the length, and w is the width.

We want to find the area of the pen, given its width. To do this, we need to express the length in terms of the width. We can use the equation we derived earlier:

50 = l + w

We can solve for l by subtracting w from both sides:

l = 50 - w

Now, we can substitute this expression for l into the area formula:

A = (50 - w)w

Expanding the right-hand side, we get:

A = 50w - w^2

This is the quadratic equation that gives the area of the pen, given its width.

Let's compare our derived equation with the options given:

A. A(w) = w^2 - 50w B. A(w) = w^2 - 100w C. A(w) = 50w - w^2

Our derived equation matches option C. This makes sense, since we derived the equation by expressing the length in terms of the width and substituting it into the area formula.

In this article, we used quadratic equations to model the relationship between the width and the area of a rectangular pen. We derived the equation A(w) = 50w - w^2, which gives the area of the pen, given its width. This equation matches option C, which is the correct answer.

This problem illustrates the importance of using mathematical models to solve real-world problems. By expressing the length in terms of the width and substituting it into the area formula, we were able to derive the quadratic equation that gives the area of the pen, given its width. This equation can be used to optimize the design of the pen and maximize its area.

  • The perimeter of a rectangle is given by the formula P = 2l + 2w.
  • The area of a rectangle is given by the formula A = lw.
  • The quadratic equation A(w) = 50w - w^2 gives the area of the pen, given its width.
  • This equation can be used to optimize the design of the pen and maximize its area.
  • [1] "Mathematics for the Nonmathematician" by Morris Kline
  • [2] "Calculus" by Michael Spivak
  • Perimeter: The distance around a shape.
  • Area: The amount of space inside a shape.
  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
    A Farmer's Fencing Problem: Q&A

In our previous article, we explored the quadratic equation that gives the area of a rectangular pen, given its width. We derived the equation A(w) = 50w - w^2 and compared it with the options given. In this article, we will answer some frequently asked questions about the problem and provide additional insights.

A: The perimeter of the pen is the total length of the fencing used to enclose it. In this problem, we are given 100 meters of fencing, which means the perimeter of the pen is 100 meters.

A: We started by using the formula for the perimeter of a rectangle: P = 2l + 2w. We set up the equation 100 = 2l + 2w and simplified it to 50 = l + w. We then solved for l by subtracting w from both sides: l = 50 - w. Finally, we substituted this expression for l into the area formula: A = lw.

A: The area of the pen is given by the equation A(w) = 50w - w^2. This equation shows that the area of the pen is a quadratic function of the width. As the width increases, the area of the pen increases, but at a decreasing rate.

A: To optimize the design of the pen, you can use the equation A(w) = 50w - w^2 to find the maximum area of the pen. To do this, you can take the derivative of the equation with respect to w and set it equal to zero. This will give you the critical point(s) of the function, which correspond to the maximum area.

A: This problem has many real-world applications in agriculture, construction, and engineering. For example, farmers may use this equation to design optimal pens for their livestock, while construction companies may use it to design buildings with maximum area.

A: Yes, this equation can be used to solve other problems involving quadratic functions. For example, you can use it to find the maximum or minimum value of a quadratic function, or to solve systems of quadratic equations.

A: Some common mistakes to avoid when solving this problem include:

  • Not simplifying the equation 100 = 2l + 2w correctly
  • Not solving for l correctly
  • Not substituting the expression for l into the area formula correctly
  • Not taking the derivative of the equation with respect to w correctly

In this article, we answered some frequently asked questions about the problem and provided additional insights. We hope this article has been helpful in understanding the quadratic equation that gives the area of a rectangular pen, given its width.

This problem illustrates the importance of using mathematical models to solve real-world problems. By expressing the length in terms of the width and substituting it into the area formula, we were able to derive the quadratic equation that gives the area of the pen, given its width. This equation can be used to optimize the design of the pen and maximize its area.

  • The perimeter of a rectangle is given by the formula P = 2l + 2w.
  • The area of a rectangle is given by the formula A = lw.
  • The quadratic equation A(w) = 50w - w^2 gives the area of the pen, given its width.
  • This equation can be used to optimize the design of the pen and maximize its area.
  • [1] "Mathematics for the Nonmathematician" by Morris Kline
  • [2] "Calculus" by Michael Spivak
  • Perimeter: The distance around a shape.
  • Area: The amount of space inside a shape.
  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.