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.

1 Answers

Best 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+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