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NeTakaya
3 years ago
13

A farmer is building a fence to enclose a rectangular area against an existing wall, shown in the figure below. A rectangle labe

led, Fenced in Region, is adjacent to a rectangle representing a wall. A rectangle labeled, Fenced in Region, is adjacent to a rectangle representing a wall. Three of the sides will require fencing and the fourth wall already exists. If the farmer has 184 feet of fencing, what are the dimensions of the region with the largest area?
Mathematics
1 answer:
Trava [24]3 years ago
3 0

Answer:

  46 ft by 92 ft

Step-by-step explanation:

The largest area is enclosed when half the fence is used parallel to the wall and the other half is used for the two ends of the fenced area perpendicular to the wall. Half the fence is 184 ft/2 = 92 ft. Half that is used for each end of the enclosure. So, the dimensions are 46 ft by 92 ft.

_____

<em>Working</em>

Let x represent the dimension parallel to the wall. Then the other dimension is ...

  (184 -x)/2

and the total area is ...

  Area = x(184 -x)/2

This is the equation of a parabola that opens downward. It has zeros (x-intercepts) at x=0 and x=184. The vertex (maximum area) is located on the line of symmetry, halfway between those x-intercepts. The maximum area is found where x=184/2 = 92.

Then the other dimension is (184-92)/2 = 46.

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