Question

An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 4 dollars per square foot and the cost to construct the four sides is 3 dollars per square foot, determine the dimensions for a box to have volume = 71 cubic feet which would minimize the cost of construction.

Answer 1

Which part of the problem is giving you trouble?

Answer 2

Well, I thought that I'd figured it out.. but when I put my answer into the computer program, it says I'm wrong. lol What I did was create an equation for the surface area. I used the volume for equation to write height in terms of width, then I substituted 4w in for lenth in the s.a. equation as well as 71/(4w^2) for the height in the surface area equation. After that I found the derivative, set it to zero and solved. I think I made a dumb mistake somewhere, so I asked on here to see if someone would just resolve for me, because I'm not catching it.

Answer 3

You have that the base length is 4 times larger than the width so you can eliminate one or the other. You have an equation for the volume, and can use that to find height in terms of the variable you kept from the first equation..

Then you can rewrite your cost function in terms of just one of the variables and take the derivative to find where it has a minimum.

Answer 4

thats what I tried to do.. lol, I'm pretty sure my math is just wrong somewhere.

Answer 5

\(L=4W\)
\(LWH=71 \implies 4W^2H = 71\implies H = \frac{71}{4W^2}\)
\(Cost = 4LW + 3(2LH) + 3(2(WH))\)
\(= 16W^2 + \frac{6(4W)(71)}{4W^2} + \frac{6W(71)}{4W^2}\)
etc.

Answer 6

Right. That's what I did. Could you just continue the algebra to simplify it? Once its simplified I shoouulld be fine taking the derivative and setting it to zero and all that fun.

Answer 7

Is that the same cost function you got?

Answer 8

I had written substituted in the 4w for the length and simplified before I put in the substitution for height.. so I had 2wh +2h(4w) +2w(4w), and then made it 10wh +8w^2 before I put in the 71/(4w^2).

Answer 9

I don't know if that would have made a difference though.

Answer 10

That looks like a surface area function though, not the cost function. You forgot to factor in the costs of each of the sides.

Answer 11

ooohhh. I didn't even realize I had left that out.

Answer 12

Simplifying we just end up with
\(Cost =16w^2 + \frac{1065}{2w}\)

Answer 13

so our derivative would be.. 32 w- 1065/(2^w2)

Answer 14

Okay I've got it. Thank you :)

Answer 15

Just to check myself did you come out with:
width=2.5
Length = 10
height =2.84