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The length of the rectangular box or dilivery box must be 5 inches, if the volume of the rectangular box is 360 cubic inches.
Let the length of the rectangular box be x inches.
According to the given question.
The volume of the rectangular box is 360 cubic inches.
The width of the delivery box is 3 inches longer than the length.
⇒ Width = 3 + x
And, the height is 4 inches longer than the length.
⇒ height = x + 4
As, we know that the volume of the rectangular box is the product of its length, width and height.
Therefore, the length of the rectangular box is given by
360 = x(x+3)(x+4)
⇒ 360 = x(x^2 + 4x + 3x + 12)
⇒ 360 = x^3 + 4x^2 + 3x^2 + 12x
⇒ 360 = x^3 + 7x^2 + 12x
⇒ x^3 + 7x^2 + 12x -360 = 0
We would apply the remainder theorem to solve the polynomial.
According to the remainder theorem, if a polynomial P(x) is divided by (x - r) and there is a remainder R; then P(r) = R.
Since,
if we take x = 5
Then the above polynomial will give 0 value.
Which means 5 is one of the root of the above polynomial x^3 + 7x^2 + 12x -360 = 0.
Therefore, the length of the rectangular box or dilivery box must be 5 inches, if the volume of the rectangular box is 360 cubic inches.
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