Given that the stage manager of a school play creates a rectangular acting area of 42 square yards.
Let the length of the rectangular acting area be x, then the width is given by 42 / x.
The number of yards of string lights that the manager need to enclose the area is given by the perimeter of the rectangular area.
Recall that the perimeter of a rectangle is given by
P = 2(length + width) = 2(x + 42/x) = 2x + 84/x
The perimeter is minimum when the differenciation of 2x + 84/x is equal to 0.
i.e. [tex]2 - \frac{84}{x^2}=0\\ \\ \Rightarrow2x^2-84=0\\ \\ \Rightarrow x^2=42\\ \\ \Rightarrow x\approx6.48[/tex]
Therefore, the minimum number of yards of string lights the manager need to enclose this area is given by
[tex]2(6.48)+ \frac{84}{6.48} =12.96+12.96=25.92\approx26\ yards[/tex]
