Question

You have 60 feet of fencing to fence in part of w w l house your backyard for your dog. you want to make sure that your dog has 400 square feet of space to run around in. the back of your house will be used as one side of the enclosure as shown.
a. write equations in terms of l and w for the amount of fencing and the area of the enclosure.
b. use substitution to solve the system of equations from part (a). what are the possible lengths and widths of the enclosure?

Answer 1

Part (a):
We have length of fence is l (side parallel to the house) and width of fence (side perpendicular to the house) is w.
We are given that:
The total fencing is 60 ft. This means that the perimeter of the yard is 60 feet. Note that the perimeter of the yard would be l + 2w not 2l + 2w. This is because on of the length is eliminated as this length is already occupied by the house. This means that:
l + 2w = 60
We are also given that:
Area of yard is 400 ft². This means that:
lw = 400
Based on the above, the system of equations describing the situation would be:
l + 2w = 60 .............> equation I
lw = 400 ...............> equation II

Part (b):
We want to get the length and width. This means that we will solve the system of equations in part a.
In equation I, we have:
l + 2w = 60
This equation can be rewritten as:
l = 60 - 2w ............> equation III
Substitute with equation III in equation II and solve for w as follows:
lw = 400
(60-2w)(w) = 400
60w - 2w² = 400
2w² - 60w + 400 = 0
(w-20)(w-10) = 0
either w-20 = 0 ...........> w = 20 ft
or w-10 = 0 ............> w = 10 ft
Now, we will substitute with the value of w in equation III to get l as follows:
at w = 20:
l = 60 - 2w = 60 - 2(20) = 60 - 40 = 20 ft
at w = 10:
l = 60 - 2w = 60 - 2(10) = 60 - 20 = 40 ft
Based on the above, we have two possibilities for the dimensions:
either length is 20 ft and width is 20 ft
or length is 40 ft and width is 10 ft

Hope this helps :)