Exercise 1.1.71

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x .

Answers

After cutting the rectangular cardboard, its width and length increased by 2 x . That is why the width of the constructed rectangular box is equal to 12 − 2 x and its length is 20 − 2 x . From the scheme we can conclude that the height of the box is equal to x .

V ( x ) = x ( 12 − 2 x ) ( 20 − 2 x ) = x ⋅ 2 ( 6 − x ) ⋅ 2 ( 10 − x ) = 4 x ( 6 − x ) ( 10 − x ) = 4 x ( x 2 − 16 x + 60 ) = 4 x 3 − 64 x 2 + 240 x .

Thus, we can define V as the function of x . Since x must be positive and bigger than 6 (if x is smaller than or equal to 6, V would be not positive), we have:

V : ( 0 , 6 ) → ( 0 , + ∞ ) , V ( x ) = 4 x 3 − 64 x 2 + 240 x .

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2023-07-23 04:19
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