The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

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Camila Mccann · Accepted Answer

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+3ts=10−3t4Total area, A= area of square,&amp;nbsp;s2, plus area of equilateral triangle, half times base (t) times height&amp;nbsp;(t&amp;nbsp;sin⁡60)A=s2+12⋅t⋅(32t)Substitute value of s found in step 1A=(10−3t4)2+34t2DifferentiatedAdt=-30+9t+43Put 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=1.88When t=1.88, s=1.09Result:s=1.09 and t=1.88

The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

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