Question

Katherine wants to construct a small box with a volume of 20 cubic inches with the following specifications. The length of a box is five more than its width. Its depth is one less than its width. What are the dimensions of the box in simplest radical form and rounded to the nearest hundredth? Only and algebraic solution will receive full credit.

Answer 1

V=LWH=20

L is 5 more than W
V=5+W

H is 1 less than W
H=-1+W

easy, sub 5+W for L and -1+W for H in LWH=V=20

20=(5+W)(W)(-1+W)
expand
20=W^3+4W^2-5W
minus 20 both sides
W^3+4W^2-5W-20=0
either factor or graph to find the possible values for W
(W+4)(W^2-5)=0
set each to zero

W+4=0
W=-4
impossible, measures cannot be negative
W^2-5=0
add 5
W^2=5
sqrt both sides
W=√5

sub back

L=5+W
L=5+√5
aprox
7.24 rounded

H=-1+W
H=-1+√5
aprox
1.24 rounded
the dimentions are
√5 by (5+√5) by (-1+√5) or aprox
2.24in by 7.24in by 1.24in

Answer 2

Width =√5,length =√5+5 ,Depth =√5-1.

Explanation:

volume of box = l.w.h ,where l is length ,w is width and h is the depth of the box.

It is given l= w+5

h= w-1.

Substituting l and h values we have

V= (w+5).w.(w-1)

Or 20 = w(w+5)(w-1) [ volume is 20 cubic in)

`20 =w(w^(2) +4w-5)`

`20=w^(3) +4w^(2) -5w`

or,
`w^(3) +4w^(2) -5w-20=0`

Taking common factor of both pairs we have:

`w^(2) (w+4)-5(w+4)=0`

`(w^(2) -5)((w+4)=0`

`w^(2) -5=0` or w+4=0

w=±√5 or w=-w

Length can not be negative so

w=√5

l=w+5=√5+5

h=w-1=√5-1