For this case, the first thing we must do is define variables.
We have then:
x: length of the garden
y: width of the garden
The garden area is:
[tex] xy = 5000 [/tex]
The perimeter of the garden is:
[tex] 2x + 2y = 300 [/tex]
Rewriting the perimeter:
[tex] x + y = 150 [/tex]
[tex] y = 150-x [/tex]
Substituting in the area we have:
[tex] x (150-x) = 5000 [/tex]
[tex] 150x-x ^ 2 = 5000 [/tex]
Completing squares:
[tex] x ^ 2-150x = -5000 [/tex]
[tex] x ^ 2-150x + (- 75) ^ 2 = -5000 + (- 75) ^ 2 [/tex]
[tex] (x-75) ^ 2 = 625 [/tex]
We discard the negative root:
[tex] (x-75) = \sqrt{625} [/tex]
[tex] (x-75) = 25 [/tex]
[tex] x = 25 + 75 [/tex]
[tex] x = 100 [/tex]
Then, the other dimension is:
[tex] y = 150-x [/tex]
[tex] y = 150-100 [/tex]
[tex] y = 50 [/tex]
Answer:
The dimensions of the garden are:
[tex] x = 100 [/tex]
[tex] y = 50 [/tex]
