Let s be the length of a side of the square, t be the length of a side of the equilateral triangle, A be their total area and P be their total perimeter. Solve to find s in terms of t. P=10=4s+3t s=(10-3t)/(4) Total area, A= area of square, s^(2), plus area of equilateral triangle, half * base (t) * height (t sin 60) A=s^(2)+(1)/(2)*t*((sqrt(3))/(2)t) Substitute value of s found in step 1 A=((10-3t)/(4))^(2)+(sqrt(3))/(4)t^(2) Differentiate (dA)/(dt)=-30+9t+4sqrt(3)t Put dA/dt = 0 and solve for t. The second derivative at this point is positive, so it is a minimum. Plug in that of t into the equation found in the first step. dA/dt = 0 when t=1.88 When t=1.88, s=1.09 Result: s=1.09 and t=1.88