SOLUTION: A large plywood box has a volume of 12 ft3. Its length is 4 ft greater than its height, and its width is 1 ft less than its height. What are the dimensions of the box?

Question 644161: A large plywood box has a volume of 12 ft3. Its length is 4 ft greater than its height, and its width is 1 ft less than its height. What are the dimensions of the box?
Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
Use the formula
(1) V = lwh
with
(2) l = h + 4
(3) w = h - 1
to obtain
(4) (h + 4 )(h - 1)h = 12
Simplify to
(5) (h^2 + 3h - 4)h - 12 = 0 or
(6) h^3 + 3h^2 - 4h - 12 = 0
Since all cubic equations have one real root, find it first by substituting different values of h into (6) until (6) = 0.
I start at h=0, then
f(0) = -12
f(1) = 1+3-4-12 = -12
f(2) = 8+12-8-12 = 0, STOP
Therefore h = 2 is a root or
(7) (h-2) is a factor of (6).
To find the other roots divide (6) by (7) to obtain
(8) h^2+5h+6 = 0
Factor (8) to obtain
(9) (h+2)(h+3) = 0
which gives the roots
(10) h = {-2,-3}
Since both are negative, they cannot be the height of the box.
Our answer is
(11) h = 2,
and from (2) and (3) we get
(12) l = 6 and
(13) w = 1
Is (lwh = 12)?
Is (6*1*2 = 12)
Is ( 12 = 12)? Yes
Answer: The box is 6ft by 1ft by 2ft.
We can also check our factoring of (6) by FOILing
(14) (h-2)(h+2)(h+3) =
(15) (h^2 - 4)(h+3) =
(16) h^3 - 4h + 3h^2 - 12 =
(17) h^3 + 3h^2 - 4h - 12
Which is the same as (6).
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