Question

The stage manager of a school play creates a rectangular acting area of 42 square yards. String lights will outline the acting area. To the nearest whole number, how many yards of string lights does the manager need to enclose this area?

Answer 1

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. $2 - \frac{84}{x^2}=0\\ \\ \Rightarrow2x^2-84=0\\ \\ \Rightarrow x^2=42\\ \\ \Rightarrow x\approx6.48$

Therefore, the minimum number of yards of string lights the manager need to enclose this area is given by
$2(6.48)+ \frac{84}{6.48} =12.96+12.96=25.92\approx26\ yards$