Perimeter of a rectangular playing field measures 1,500 feet. Find the maximum area of the field.
Let the length of the field be l and width w. Then, 1,500 = 2(l + w), meaning that 750 = l + w. This means that w = 750 - l, and since the area is given by A = lw, we have
A = l(750 - l)
= -l2 + 750l
= -(l2 – 750l)
= -[l2 – 750l + (-750/2)2 – (-750/2)2]
= -(l2 – 750l + 140,625) + 140,625
= -(l – 375)2 + 140,625.
This is a vertex form of a parabola. Since there is a negative coefficient in front of l, the parabola expands downward, and the vertex represents its maximum value. The vertex is given by (375, 140625), thus for the length 375 feet, the maximum area of the playing field is 140,625 square feet with the given perimeter of the field. Note that you could have expressed the area in terms of the width w, and it would not have changed the result.
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