Helps with Calculus Optimization Word Problem?

Question

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

Any tips on optimization problems would be appreciated.

Answer 1

if x is the side of the square and y is the side of the triangle, then 4x + 3y = 10

y = (10 - 4x)/3

The area of the square is x²

The area of the triangle is side times height over 2
then this is y * [y sqrt(3) / 2] / 2 = y² sqrt(3) / 4

The total area is f(x) = x² + y² sqrt(3) / 4

f(x) = x² + (10 - 4x)² sqrt(3) / 36
= x² + (16x² - 80x + 100) sqrt(3) / 36
= x² + (4x² - 20x + 25)sqrt(3) / 9

then f '(x) = 2x + (8x - 20) sqrt(3) / 9
=2x + 8 x sqrt(3) / 9 - 20 sqrt(3) / 9

f '(x) = 0 if x = [20 sqrt(3) / 9] / [2 + 8 sqrt(3) / 9]
=20 sqrt(3) / [18 + 8sqrt(3)]
= 10 sqrt(3) / [9 + 4sqrt(3)]
= 10 / [4 + 3sqrt(3)]

the area is minimum for this value of x.

Then y = (10 - 4x) / 3
y = (10 - 40 / [4 + 3sqrt(3)]) / 3
= (40 + 30sqrt(3) - 40) / (3[4 + 3sqrt(3)])
= 10sqrt(3) / [4 + 3sqrt(3)]

Answer 2

Let T=side of triangle and S=side of square

Perimeter = 10 = 3T + 4S; S = (10-3T)/4

Area A = (T^2)(SQRT3)/2 + S^2
= (T^2)(SQRT3)/2 + [(10-3T)/4]^2.

Find dA/dT, set equal to 0, solve for T and S, check if area is min