A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is

Step-by-step solution

Let the side of square be 22 - l 4 and side of triangle be l 3 , So, total area is given by Δ = 3 4 l 3 2 + 22 - l 4 2 ⇒ Δ = 1 12 3 · l 2 + 1 16 22 - l 2 Differentiating with respect to l to get maxima and minima points, ⇒ d Δ d l = 1 6 3 · l - 1 8 22 - l = 0 ⇒ l = 66 3 4 + 3 3 Also d 2 Δ d l 2 = 1 6 3 + 1 8 ( + ve . which is case of minima) So, side of triangle will be = l 3 = 22 3 4 + 3 3 = 66 4 3 + 9