You have just bought a new puppy and want to fence in an area in the backyard for her to roam. You buy 100 linear feet from Home Depot and have decided to make a rectangular fenced in area using the back of your house as a side. What are the dimensions to maximize the area your puppy can roam
Question
Answer:
50 ft * 25 ft
Step-by-step explanation:
Let the length of the rectangular area be x ft and the width of the rectangular area be y ft. You already have 100 ft of material for fencing. Also one side of your house is used in fencing.
The perimeter of the fence needed = x + 2y
Therefore:
100 = x + 2y
x = 100 – 2y
Also the area of the rectangular fence is:
Area (A) = length * breadth
A = xy
Substitute x = 100 – 2y
A = (100 – 2y)y
A = 100y – 2y²
Maximum area is at dA / dy = 0. Hence:
dA/dy = 100 – 4y
100 – 4y = 0
4y = 100
y = 25 ft
substitute value of y in x:
x = 100 – 2y = 100 – 2(25) = 100 – 50 = 50 ft
x = 50 ft
The dimension of the rectangular space is 50 ft * 25 ft