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 :)
