SOLUTION: Packaging is one important feature in producing quality products. A box designer needs to produce a package for a product in the shapes of the pyramid with a square base having a t

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Question 1187070: Packaging is one important feature in producing quality products. A box designer needs to produce a package for a product in the shapes of the pyramid with a square base having a total volume of 200 cubic inches. The height of the package must be 4 inches less than the length of the base. Find the dimension of the product
Answer by ikleyn(52858) About Me  (Show Source):
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Packaging is one important feature in producing quality products. A box designer needs to produce a package
for a product in the shapes of the pyramid with a square base having a total volume of 200 cubic inches.
The height of the package must be 4 inches less than the length of the base. Find the dimension of the product
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The "volume" equation is

    %281%2F3%29%2Ax%5E2%2A%28x-4%29 = 200   cubic inches,

or

    x%5E2%2A%28x-4%29 = 600.



One solution can be easily guessed:  x = 10.


Indeed,  10%5E2%2A%2810-4%29 = 10%5E2%2A6 = 600.


Next, notice that the function x%5E2%2A%28x-4%29  monotonically increases in its domain x > 4.


It means that x = 10  is the UNIQUE solution to the "volume" equation, and THERE IS NO other real solution.


ANSWER.  Under given conditions, the unique set of the dimensions of the pyramid is  10 inches for the square base 
         and  6 inches for the height.

Solved.