Mike's family wants to build a rectangular fenced backyard area for their dog. They have a 20-meter length of wire fence and four posts. They can also use the 20-meter straight length of the back of their house as a side of the enclosure, but the fence cannot attach directly to the house. The fence must stretch taught between posts, and they have fasteners to attach the fence to the posts. Describe or sketch a design for the dog enclosure that yields the maximum area using the resources they have. List the dimensions of the enclosure and list the total area. Check your work by describing a similar design that does not enclose as much area.
the conditions require that the 4 posts are the corners of the rectangle. So, if the width is x, and the length is along the house, the fence used is x+x+(20-2x)
area = x(20-2x) = 2x(10-x)
the max is achieved when x is midway between the roots, or x=5.
The pen is thus 5x10, with area = 50
suppose the width is 6. The pen is then 6x8 with area = 48
suppose the width is 4. Then the pen is 4x12 with area = 48.
To find the design that maximizes the area of the dog enclosure, we need to consider the given constraints and think about how to distribute the 20-meter length of wire fence and four posts most effectively.
Design for Maximum Area:
1. Place two posts at the ends of the 20-meter length. This forms the width of the enclosure.
2. Attach the wire fence to these two posts, creating a straight line.
3. Place the remaining two posts at the midpoint of the wire fence, dividing it into two equal halves.
4. Attach the wire fence to these two posts, forming the two shorter sides of the rectangular enclosure.
In this design, the enclosure consists of a rectangular shape with one side formed by the back of the house. Let's calculate the dimensions and the total area:
Length of the enclosure (excluding the house side) = 20 meters - width of the house
Width of the enclosure = half the length of the wire fence between the two middle posts.
To validate this design, consider a scenario where the width of the enclosure is equal to the length of the house side:
Length of the enclosure = 20 meters - width of the house
Width of the enclosure = half the length of the wire fence between the two middle posts.
Design with Less Enclosed Area:
In this case, let's assume the width of the enclosure is less than half the length of the wire fence between the two middle posts.
By comparing the two designs, it becomes clear that the design providing the maximum enclosed area is the one described above, where the width of the enclosure is equal to half the length of the wire fence between the two middle posts.
Please note that the specific dimensions of the enclosure and the area enclosed will depend on the exact measurements of the house width and the wire fence between the middle posts.