A piece of paper of dimensions L × W, L > W inust be cut to form a box of dimens

Question

A piece of paper of dimensions L × W, L > W inust be cut to form a box of dimensions l × w × h as showrn in Figure 1. Figure 1: Diagram for Question 2. (a) Give an expression for and w in ters of L, W and h (b) Using the expressions above, give an expression for the box volume v- (c) What limitations must be applied to h in terms of L and/or W to ensure d) Find the value of h that results in the box with the maximum volume, lwh in terms of L, W and h. the box is physical? h* in terms of L and W. You do not have to calculate the corresponding volume for the general case.

Explanation / Answer

Meaning of Question: You want to convert a piece of paper with dimension LXW into a box which has a height of h

(a)

The new length, l = L - 2h (h deduction from both sides of length)

w = W - 2h (h deduction from both sides of width)

(b)

Volume of box = length * width * h = l * w * h

=> (L - 2h) * (W - 2h) * h

=> (Lh - 2h^2) * (W - 2h)

=> WLh - 2Lh^2 - 2Wh^2 + 4h^3

(c)

Now for the box to be physical, all the dimensions must be positive, so we get the constraints

h > 0

w > 0

l > 0

W - 2h > 0

L - 2h > 0

Now, since L > W, hence the limitation will come from W condition, so we must have

W - 2h > 0 or h < W/2

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