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The first problem asks to find the minimum cost of a rectangular box if the bottom costs \$2, the sides \$5 per square foot, and the top costs $7 per square foot. The volume of the box is given as 20 cubic feet. I think I found the answer to be that all of the sides will be 2.714, but then it asks to "demonstrate that you have found the minimum using the discriminant, D" which I don't understand. Isn't the discriminant D=b^2-4ab?

The next question asks to make a cylindrical vase (cylinder with no top) where the bottom is 0.3cm thick and the sides are 0.2cm thick. The vase needs to hold 1000 cubic cm. It then asks what the dimensions should be to minimize the weight. We are supposed to use Lagrange Multipliers, but I got that the height was something over zero.

Any help would be greatly appreciated!

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You've posed two questions. Perhaps we'll give you a solution for the first one and leave the other as an exercise? Fair?

Let a, b and c be length, width and height.

We write $F(a,b,c)=(2+7)ab+5c(a+b)=9ab+100\frac{a+b}{ab} $. Then:

$\\ \frac{\partial F}{\partial a} = 9b - 100/a^{2} \equiv 0 \\ \frac{\partial F}{\partial b} = 9a -100/b^{2} \equiv 0.$

This reduces to $a^2b-b^2a = ab(b-a) = 0$ and hence (since we excluded both the possibilities of $a=0$ and $b=0$ in this algebraic step), $a=b$.

Now, $a^2b = a^3 = 100/9$ and so $a=b=(\frac{10}{3})^{2/3}$. Now $c=20\cdot (\frac{10}{3})^{-4/3}=6\cdot (\frac{3}{10})^{1/3}.$

Now, you must check whether it is a minimum or maximum: just take the determinant of the Hessian and you'll get $(200/a^3)^2-81 = 18^2 - 81 > 0$ and so this is a local minimum. Now just use the continuity of $F$ and the fact that the place where $\nabla F =0$ is unique to conclude that this is a global minimum.

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