I don't remember the whole question, since it was part of an exam that I just finished.
It is an optimization problem for building a rectangular snap together sandbox with an open top.
The sandbox should hold 32 cubic feet of sand, and you are supposed to minimize the materials used to build the sandbox.
It asks to find Length, width and height
In general, you can use the method of Lagrange Multipliers to solve multivariate calculus optimization problems. This method involves solving the following system of equations:
$$\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$$ $$g(x,y,z) = k$$
First, you want to break down the first equation into its constituent parts. Since we are working with vectors, we know that the components must be equal in order for the two vectors to be equal:
$$f_x = \lambda g_x$$ $$f_y = \lambda g_y$$ $$f_z = \lambda g_z$$
Next, we want to ask ourselves, "What is the problem asking?" Based on what you wrote, it seems like the following would be a powerful way to analyze the question: "Find the dimensions of the open-topped rectangular prism with the smallest surface area, given that the total volume is 32 cubic feet." By phrasing the question in this manner, we're able to derive the following information from it to use within our problem analysis:
$$f(x,y,z) = xy + 2xz + 2yz$$ $$g(x,y,z) = xyz$$
In general, an effective strategy is to set the function F equal to the property of the system that you are looking for, and the function G equal to the property of the system that you already know. I think you could still complete the problem if you take the opposite approach, but I wouldn't risk it. In this example, we're looking to minimize materials used, so we set the surface area formula equal to F. We already know the volume, so we set the volume formula equal to G. Now, we can rewrite our system of equations:
$$y + 2z = \lambda yz$$ $$x + 2z = \lambda xz$$ $$x + 2y = \lambda xy$$ $$xyz = 32$$
In this particular case, notice how we can multiply Equation (1) by x, Equation (2) by y, and Equation (3) by z, and we are left with a startlingly simple system of equations? Observe:
$$x(y + 2z) = \lambda xyz$$ $$y(x + 2z) = \lambda xyz$$ $$z(x + 2y) = \lambda xyz$$
From there, we can solve for x, y, and z quite easily:
$$(1) = (2): xy + 2xz = xy + 2yz$$ $$\Rightarrow 2xz = 2yz$$ $$\therefore x = y$$ $$(2) = (3): xy + 2yz = xz + 2yz$$ $$\Rightarrow xy = xz$$ $$\therefore y = z$$
Now we know that x, y, and z are all the same. We plug this into the constraint, to get the values for each. We only need to do this once since we already know they're equal:
$$xyz = 32$$ $$\Rightarrow x^3 = 32$$ $$\therefore x = y = z = 2 \sqrt[3]{4}$$
Now, all that's left is to find the surface area by plugging the values for x, y, and z back into your function F:
$$f(2 \sqrt[3]{4},2 \sqrt[3]{4},2 \sqrt[3]{4}) = 2 \sqrt[3]{4} * 2 \sqrt[3]{4} + 2 * 2 \sqrt[3]{4} * 2 \sqrt[3]{4} + 2 * 2 \sqrt[3]{4} * 2 \sqrt[3]{4}$$ $$= (2 \sqrt[3]{4})^2 + 2(2 \sqrt[3]{4})^2 + 2(2 \sqrt[3]{4})^2$$ $$= 5(2 \sqrt[3]{4})^2$$ $$= 20(\sqrt[3]{4})^2$$ $$= 40 \sqrt[3]{2}$$
That should be the minimum amount of required materials, in square feet.