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A candy factory needs a box that has a volume of 30 cubic inches. The width should be 2 inches less than the height and the

length should be 5 inches greater than the height. What should the dimensions of the box be?

A candy factory needs a box that has a volume of 30 cubic inches. The width should-example-1
User Griwes
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1 Answer

10 votes
Volume is length*width*height
V = LWH

Height = H
Width = H-2
Length = H+5

V = H(H-2)(H+5) = 30

H(H2+3H-10)=30
H3+3H2-10H = 30
H3+3H2-10H-30 = 0

Per the rational roots theorem the roots of the
polynomial will be within the factors of the
constant , -30, divided by the coefficient of
the highest term, 1 from H3
The real roots must be within the factors of -30/1
or the factors of -30
Factors of -30 are ± {1, 2, 3, 5, 6, 10, 15, 30}

We will use synthetic division to determine the roots.

I first tried 1, -1, 2, -2, 3, and determined that -3 is
a root of the polynomial H3+3H2-10H-30.
This means that (H+3) is a factor ... Note that if
H=-3 then H+3 = 0

The synthetic division looks like this:

-3 | 1 3 -10 -30 the root and the coefficients
| -3 0 30
---------------------------
1 0 -10 0

Since the last sum is 0, there is no remainder.
That means that H=-3 is a root of the polynomial.
The other terms from right to left become the
constant and coefficients of the new polynomial
in ascending order ... LEFT to RIGHT

H3+3H2-10H-30 = (H+3)(H2+0H-10)

(H+3)(H2-10) = 0

Since H=-3 will not work for positive height we
need to look at:

H2-10 = 0
H2 = 10
H = ±√10

Again, discarding the negative

Height = √10 inches

Width = H-2 = -2 + √10 inches

Length = H+5 = 5 + √10 inches
User Yahya Hussein
by
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