Answer: 274/3 and 137 ft
Step-by-step explanation:
Total fence used: 548 ft
So, perimeter will be 548 ft, including the fence that divides the land.
Consider the playground the drawing below.
__________x________
| |
| | y
| |
|__________________|
Let's consider that the playground is going to be divided parallel to y (and there's no problem, because if you choose parallel to x, the area will be the same). So, the perimeter (the sum of all sides) will be:
2x + 3y = 548
And the area
A = x.y
Isolating x, we have:
x = (548 - 3y)/2
Substituting the x in the Area equation:
A = (548 - 3y)/2 . y = 548y - 3y²/2 = 274y - 3/2y²
A = 274y - 3/2y²
So, it's a quadratic function. And, to find out the maximum area that the playground can have, we need to find the y-vertex of the parabola: y-vertex = Δ/4a
a = -3/2 b = 274 c = 0
Δ = 274² - 4.(-3/2).0 = 274² = 75076
Amax = -Δ/4a = -75076/4.(-3/2) = 75076/6 = 12512.7 ft²
y = -b/2a = -274/2.(-3/2) = 274/3 ft
x = (548 - 3y)/2 = (548 - 3.274/3)/2 = 274/2 = 137 ft
