Solve the problem
Packaging is one important feature in producing quality products. A box designer needs to produce a package for a product in tge shape of a pyramid with absquare base having a total volume of 36 cubic inches. The height of the package must be 4 inches less than the length of the base. Find the dimensions of the product.
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Answer:
The volume of a pyramid with a square base is given by the formula: V = (B^2 * H) / 3, where B is the length of the base and H is the height of the pyramid.
Given that the volume of the pyramid is 36 cubic inches, we can set up the equation:
(B^2 * (B - 4)) / 3 = 36
Expanding and simplifying the equation, we get:
B^3 - 4B^2 = 108
B^3 - 4B^2 - 108 = 0
This is a cubic equation, which can be solved using various methods such as factoring, the rational root theorem, or numerical methods.
One possible method to solve the equation is to use synthetic division. We can use a guess of B = 6 as the rational root, since 6 is a factor of both 108 and 6.
Synthetic division gives us:
6 | 1 -4 -108
| 6 -24 -648
| -18 -60 540
| -78 480
| 402
The remainder is 402, which means that 6 is not a root of the equation. However, we can try dividing by another factor of 6, which is -1.
Synthetic division gives us:
-6 | 1 -4 -108
| -5 -10 102
| 5 -100
| -95
The remainder is -95, which means that -6 is a root of the equation. We can write the factorization:
B^3 - 4B^2 - 108 = (B + 6)(B^2 - 10B + 18)
Since the length of the base must be positive, B must be greater than 6. Therefore, the only solution for B is 10.
The length of the base of the package is 10 inches, and the height of the package is 10 - 4 = 6 inches.