Area problem, involving quadratic functions

In summary, the Cattle farmer wants to build a rectangular enclosure that has a maximum area using the least amount of fencing possible.
  • #1
bonnieerika
4
0
Hi, I don't understand what this question is asking and I have idea how to do it.. any help is very much appreciated! I understand how to complete the square, parabolas and such and the concept of maximum and minimum, I just don't understand this question.

A Cattle farmer wants to build a rectangular fenced enclosure divided into 5 rectangular pens. Each pen has equal area.

[diagram shows picture of 5 rectangular pens side by side, all of them are the same size]

A total length of 120 m of fencing material is available. Find the overall dimensions of the enclosure that will make the total area a maximum.
 
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  • #2
bonnieerika said:
Hi, I don't understand what this question is asking and I have idea how to do it.. any help is very much appreciated! I understand how to complete the square, parabolas and such and the concept of maximum and minimum, I just don't understand this question.

A Cattle farmer wants to build a rectangular fenced enclosure divided into 5 rectangular pens. Each pen has equal area.

[diagram shows picture of 5 rectangular pens side by side, all of them are the same size]

A total length of 120 m of fencing material is available. Find the overall dimensions of the enclosure that will make the total area a maximum.

Start by drawing out the enclosure and labelling the dimensions of a single rectangle with x and y (or you can give the entire enclosure those variables if you wish).
Now, what is the area equal to in terms of x and y? i.e. Find A(x,y).
What about the length of fencing in terms of x and y? We also know that the fencing is 120m so you'll have an equation in two variables given by F(x,y)=120.
 
  • #3
bonnieerika said:
Hi, I don't understand what this question is asking and I have idea how to do it.. any help is very much appreciated! I understand how to complete the square, parabolas and such and the concept of maximum and minimum, I just don't understand this question.

A Cattle farmer wants to build a rectangular fenced enclosure divided into 5 rectangular pens. Each pen has equal area.

[diagram shows picture of 5 rectangular pens side by side, all of them are the same size]

A total length of 120 m of fencing material is available. Find the overall dimensions of the enclosure that will make the total area a maximum.

You really do need to make an effort before asking for help, but here are some hints.

Try drawing a couple of different "designs", each using a total of 120 of fencing and having 5 equal rectangular pens. Do you see how you can have different areas?

Now start defining some "variables" connected with your designs, and try to express two things in terms of them: (1) the total area enclosed; and (2) the total length of fencing used. Equating (2) to 120 m will give a relationship between your variables, and that will help you with the task of maximizing (1).
 

FAQ: Area problem, involving quadratic functions

What is the area problem involving quadratic functions?

The area problem involves finding the maximum or minimum value of a quadratic function, which is represented by a parabola. The maximum or minimum value corresponds to the highest or lowest point on the parabola, which is also known as the vertex. The area problem involves finding the maximum or minimum area of a shape, such as a rectangle or triangle, given certain constraints and using a quadratic function to represent the area.

How do you solve the area problem involving quadratic functions?

To solve the area problem, you first need to determine the constraints, such as the fixed perimeter or base length. Then, you can use the formula for the area of the shape, such as A = lw for a rectangle, and substitute the constraints into the formula. This will give you a quadratic function in terms of one variable, which you can then graph to find the maximum or minimum value. You can also use the quadratic formula to find the exact coordinates of the vertex.

What is the relationship between the area problem and optimization?

The area problem is a type of optimization problem, where the goal is to find the maximum or minimum value of a function. In the area problem, the function represents the area of a shape, and the maximum or minimum value corresponds to the maximum or minimum area. Therefore, finding the solution to the area problem also means optimizing the area of the shape.

What are some real-life applications of the area problem involving quadratic functions?

The area problem involving quadratic functions has many real-life applications, such as in architecture, engineering, and physics. For example, architects may use the area problem to determine the dimensions of a room with a fixed perimeter to maximize the usable space. Engineers may use it to determine the dimensions of a bridge or building to minimize material costs. In physics, the area problem can be used to calculate the trajectory of a projectile or the shape of a mirror to reflect light in a certain way.

What are some common mistakes when solving the area problem involving quadratic functions?

Some common mistakes when solving the area problem include not properly identifying the constraints, using the incorrect formula for the area of the shape, and making calculation errors. It is also important to check the units of measurement and ensure they are consistent throughout the problem. Additionally, when graphing the quadratic function, be sure to plot enough points to accurately determine the maximum or minimum value.

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