Dimensions of Box from Volume, Surface Area

[Guest]

A variation on the box problem:

A rectangular box has a square base and no top.

Volume - 18 cu. in.
Surface area - 33 sq. in.

Determine dimensions of box.

Volume = Length x Width x Height or Depth

Surface area = 2(H x W) + 2(H x L) + 2(W x L)

My partial attempt:

V = 18 = L x W x H

Surface Area = 33 = 2(H x W) + 2(H x L) + 2(W x L)

How do I solve from this point? Thanks.

[Guest]

As before get as much info from the question as you can.....
Square base....let sides of square = S .....then area of base is S^2
Let H = height .....then volume of box is (S^2) x H = 18
Surface area will be base area plus sides area
that is.....S^2 + 4 x (S x H) = 33

We have 2 unknowns and 2 equation so can now solve for S and H....simultaneous equations....

HS^2 = 18 ......(1)
S^2 + 4SH = 33 .....(2)
..............................................

[Guest]

HS^2 = 18 ......(1)
S^2 + 4SH = 33 .....(2)

S^2H = 18

S^2 + 4HS = 33

I can't solve them. Sorry.

[Guest]

what is your problem....????
Is it the simultaneous equations....use substitutions to eliminate one of the unknowns
H = 18/S^2...from .....eqn(1)
Subs this in the other equation (eqn 2) and end up with a cubic equation in "S" only. Then solve for S.

[Guest]

Sorry I posted the question.

[Guest]

HS^2 = 18 ......(1).....rearranges gives....H = 18/S^"
S^2 + 4SH = 33 .....(2)

Subs in (2) gives.... S^2 + 4xSx18/S^2 = 33
The S cancel gives...S^2 + 4x18/S = 33
multiply across by S gives S^3 + 72 = 33S
put in standard form gives S^3 - 33S + 72 = 0
This is a cubic equation, if we can a simple rational root this would be a possible answer....
The factors of the first term id S^3 is 1 and the factors of the last term is any number that divides into 72
eg. 1,2,3,4,etc up to 73
Try these in our cubic above and see if we get Zero......
WE see 3 works...so S = 3 is a possible answer
OR we can say (S - 3) is a factor of the cubic equation.
To find any others we can divide (S-3) into the cubic eqn to get a quatratic and that will give 2 more roots

But so far S = 3 therefore H = 2 from the eqns above
which answers this question of the dimensions of the box

[Guest]

Thanks.

[Guest]

I still don't understand your solution, but thanks anyway.