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A cardboard box has a square base, with each of the four edges of the base having length of x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in.
(a) Express the volume V of the box as a function of x.
(b) Since both x and V represent positive quantities (length and volume, respectively), what is the domain of V?
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure above
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| Photo by Math Principles in Everyday Life |
The length of the square base of side x is given in the problem but the height of the rectangular box is not given. Let's assign h as the height of the rectangular box.
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| Photo by Math Principles in Everyday Life |
We know that the volume of a rectangular parallelepiped is
Since the base of a rectangular box is a square, then it follows that L = W = x. The height of the rectangular box is not given in the problem. If the perimeter or the total length of the sides of a rectangular box is given, then we can get the value of h which is the height as follows
(a) Therefore, the volume of a rectangular parallelepiped is
(b) The volume of a rectangular parallelepiped can be written as
If you will examine the above equation, we can assign x = 0 up to x = 18. Obviously, we cannot have a negative value of x which is the length of the sides of a rectangular box. Also, we cannot have x > 18 because the value of V will be negative. Therefore the domain of V will be equal to
