A function assigns values. The width of the rectangle to keep the garden area maximum is 40 meters.
A function assigns the value of each element of one set to the other specific element of another set.
As it is given that the area of Simon's field is given by the function, therefore, to find the maximum of the function we need to differentiate the function and then equate it with zero.
[tex]A(x)=-x(x-80)\\\\A(x)=-x^2+80x\\\\\dfrac{dA}{dx} = -2x+80[/tex]
Now equating the differentiated function with zero we will get.
[tex]0=-2x+80\\\\-2x=-80\\\\2x=80\\\\x=40[/tex]
Thus, the width of the field should be zero.
If the width is 40 meters, then the length should also be 40 meters to keep the total fencing as 160 meters. But, if all the sides will be 40 it will be square.
As we know that a square is a form of a rectangle therefore if the width of the rectangle is kept at 40 meters, it will produce the maximum garden area.
Hence, the width of the rectangle to keep the garden area maximum is 40 meters.
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