Building volume using lagrange multipliers A rectangular building with a square front is to be constructed of materials that costs 20 dollars per square foot for the flat roof, 20 dollars per square foot for the sides and the back, and 14 dollars per square foot for the glass front. We will ignore the bottom of the building. If the volume of the building is 5,600 cubic feet, what dimensions will minimize the cost of materials? (Round your answers to the nearest hundreth such that the dimensions increase from the smallest to the largest.)
I am trying to do this problem, and i went through it twice using y as the length and x as the width and height. I tried substituting and integrating the volume and surface area formula but my answer, $11.33 \times 22.23 \times 22.23$ was wrong and I cant really figure out where to go next with this problem. 
 A: Since the building has a square front its dimensions are $x \times x \times y$ where $x$ is both the height and width of the building and $y$ is its length. The areas of the front and back are $x^2$, and the areas of the sides and roof are $xy$. The cost of the materials to construct the building is given by $$C(x,y) = 20xy + 20xy + 20xy + 20x^2 + 14 x^2 = 60xy + 34 x^2$$ by considering, in order, the roof, the two sides, the back, and the front. You wish to minimize $C(x,y)$ subject to the constraint that $$V(x,y) = x^2y = 5600.$$
The Lagrange multiplier is a number $\lambda$ satisfying $\nabla C(x,y) = \lambda \nabla V(x,y)$.  Thus you get the system of equations 
$$\begin{array}{rl} 60 y + 68 x &= 2\lambda xy \\ 60x &= \lambda x^2 \\ x^2y &= 5600.\end{array}$$
Solving nonlinear systems can be a bit ad-hoc. Since $x=0$ isn't a meaningful dimension for the building the solution requires $x \not= 0$. You can divide the middle equation by $x$ to get $$60 = \lambda x.$$ You can plug this into the first equation to obtain $$60 y + 68 x = 120 y$$ so that $68 x = 60 y$. Finally multiply the last equation by $68^2$ to get $$(68 x)^2 y = (68)^2\cdot 5600$$ so that $$60^2 y^3 = (68)^2\cdot 5600.$$
This gives you $y = 19.30$ and consequently $x = 17.03$.
