nemired9
2021-12-21
Consider the following problem:A farmer wants to fence in a rectangular area with 750 feet of fencing, then divide it into four pens by running fencing parallel to one of the sides. What is the largest possible total area of the four pens? Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, n pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so estimate it
Jenny Sheppard
Beginner2021-12-22Added 35 answers
We can devise a formula that allows us to choose a width value and obtain values for the lengths of the other sides, including the lengths that divide the interior into four pens. If the width is w then start with
Since you have 750 ft of fencing to work with, subtract 2w from it to get what is left for the other sides. 2w can not be 750 or more for the formula to work. Then divide that by 5, since there are 5 parallel sides.
length of a parallel side
As an alternative, you could create a formula where you could choose the length of the five parallel sides.
Experimenting with the numbers, this seems to be the maximum area you can get:
Wendy Boykin
Beginner2021-12-23Added 35 answers
nick1337
Expert2021-12-27Added 777 answers
For any rectangle, the one with the largest area will be the one whose dimensions are as close to 6 square as possible.
However, the dividers change the process to find this maximum somewhat.
Letting x represent two sides of the rectangle anc the 3 parallel dividers, we have 2x+3x = 5x.
Letting y represent the other two sides of the rectangle, we have 2y.
Solving for y, we first subtract 5x from each side:
Next we divide both sides by 2:
We know that the ares of a rectangle is given by
A= lw, where | is the length and w is the width. In this rectangle, one dimension is x and the other is y, making the area
Substituting the expression for y we just found above, we have
This is a quadratic equation, with values a = -2.5, b = 375 and c=0.
To find the maximum, we will find the vertex. First we find the axis of symmetry, using the equation
Substituting this back in place of every x in our area equation, we have
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Pet Planet | 110 |
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