# How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

Take the derivative of the Cost with respect to width and set it to zero to determine critical point(s).

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To minimize the total cost of materials used for the rectangular milk carton box, you need to minimize the cost function. The cost function, ( C ), is the sum of the costs of the sides and the costs of the top and bottom of the box.

[ C(w, l, h) = 2(4wl + 4lh) + 2(8wh) ]

Subject to the constraint that the volume of the box is 534 cubic cm:

[ V(w, l, h) = wlh = 534 ]

To find the dimensions that minimize the cost, you can use the method of Lagrange multipliers or solve for one variable in terms of the other two using the volume constraint and substitute it into the cost function. Then, find the critical points of the resulting function and determine which one minimizes the cost.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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