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snow_tiger [21]
3 years ago
9

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 548 fee

t of fencing is used. How do you find the dimensions of the playground that maximize the total enclosed area?
Mathematics
1 answer:
aliina [53]3 years ago
4 0

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

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