Question

A gardener has 140 feet of fencing to put around a rectangular vegetable garden. The function A(w)= 70w - w^2 gives the garden's area A (in square feet) for any width w (in feet). Does the gardener have enough fencing for the area of the garden to be 1300ft^2? Determine whether the solution of the equation is real or non-real.

Answer 1

The largest area that can be enclosed will be that of a square (140 ft)/4 = 35 ft on a side. Such a square has an area of (35 ft)² = 1225 ft², not enough for the garden to be 1300 ft². This tells us the solution to the equation you're being asked to write is non-real.

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Putting the given numbers in the area equation, you have

... 1300 = 70w -w²

Multiplying by -1 and completing the square, we have

... w² -70w +35² = -1300 +35²

... (w -35)² = 1225 -1300 = -75

Taking the square root shows us the solution is non-real.

... w - 35 = ±√(-75) = ±5i√3

... w = 35 ± 5i√3 . . . . . . . . . after adding 35. These are complex solutions.