Problem:
Suppose you are assigned to design a box for the shoe products having a total volume of 160 cubic inches. The length of the box is 4 inches longer than the height and the width is 3 inches shorter than the length. Find the dimensions of the box.
Solution:
Solving the volume we need to use the formula of volume which is: (length × width × height)
In representation,
Let x be the height
Then, let x + 4 be the length
And let (x + 4) - 3 be the width
The volume is 160 inch.
The volume being 160 inch gives us the equation.
- 160 = (x + 4)(x + 4 - 3)(x)
- 160 = (x + 4)(x + 1)(x)
- 160 = (x² + 4x)(x + 1)
- 160 = x³ + 5x² + 4x
- 160 - x³ + 5x² + 4x = 0
- -x³ - 5x² - 4x + 160 = 0
- (x - 4)(x² + 9x + 40) = 0
- x - 4 = 0 or x² + 9x + 40 = 0
- x = 4
The depressed equation is x² + 9x + 40 = 0 has no rational root, since it is not factorable Therefore, the polynomial has a root of 4. Hence, the volume of the height is 4.
If the height is 4 inch, then substitute the x in the equation of the length and the width to find its value.
If the length is x + 4 then,
- L = x + 4
- L = (4) + 4
- L = 8
If the width is (x + 4) - 3 then,
- W = (x + 4) - 3
- W = (8) + 4 - 3
- W = 8 - 3
- W = 5
Answer:
Thus, the dimension of the box are the height is 4inch, then the length is 8inch and the width is 5 inch.
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