I want to construct a rectangular storage container that has a square base, a closed top, and a volume of 9ft^3. If the material for the base costs $5/ft^2 and the material for the sides and to costs $4/ft^2, what is the cost of the cheapest such container?

Question

anonymous · Answer

What i've got so far is are of base is w*l, cost of was is 5wl
cost of front is 4lh
cost of back is 4lh
cost of left side is 4wh 
cost o right side is 4wh
.... total cost is 5wl+2(4lh)+2(4wh)
                      =5wl+8lh+8wl

anonymous · Answer

4wl+8lh+8wh (typo)

campbell_st · Answer

so you are combining volume and surface area 

if the base has dimensions of   l x l the area is l^2 

and let the height = h

the volume is   

$$V = l^2 \times h$$

and you know the volume is 9

so 

$$9 = l^2 \times h$$

this formula can be changed to make h the subject

$$h = \frac{9}{l^2}$$

the surface areas  

top and bottom  area are the same 

$$A =2 l^2$$

the 4 sides have length l and width h 

so there areas are 

$$A = 4 \times l \times h$$

So the cost of the box is 

$$C = 4(2l^2) + 5(4l \times \frac{9}{l^2})$$

so the cost can be simplified to 

$$C(l) = 8l^2 + \frac{180}{l}$$

I'd then differentiate with respect to l

$$C'(l) = 16l - \frac{180}{l^2}$$

set C'(l) = 0 

and some for l. this will give the value of l that gives the minimum cost. 

you would then need to substitute to find the height value. 

Hope this makes sense.... and you have done calculus.

anonymous · Answer

Thank you... what I don't understand is how you put it all into a formula. I understand how to label it, I just want to understand how you went about making the forumla

campbell_st · Answer

well you have the right idea with your work so far, its just the base, and lid, are squares so you are only dealing with 2 variables, length and height. 

so you are using the volume information to express combined surface area with 1 variable, length. 

For most of these types of questions, it just practice and carefully reading... 

and the key yo 1 formula is you are asked for minimum cost... 

so that has to be the equation, or formula.

anonymous · Answer

I'm sorry, but I still do not quite understand.

anonymous · Answer

Can you explain it in terms of v=l*w*h with the information that I have?

anonymous · Answer

Like the formula I made is 5wl+8lh+8wh .... what i'm asking is how could I find the length, width and heigh using the v=lwh formula?

anonymous · Answer

(i'm trying to find the domain and critical numbers)

campbell_st · Answer

ok... so the square base 

|dw:1371587189052:dw|

you have 2 squares, top and base of the box. 

the sides 

|dw:1371587254469:dw|

you have 4 of these recatangles 

so the total area of the surfaces are 

$$A = 2l^2 + 4 \times l \times h$$

now to make times simpler... you know some volume information with a square base 

the volume of a prism is 

V = Base area x height

so 
$$9 = l^2 \times h$$

making h the subject you get 

$$h = \frac{9}{l^2}$$

and lastly changing the Total surface area to a cost equation 

$4 per ft^2 for base and top  $5 for sides

$$C = 4(2l^2) + 5(4lh)$$

use the value for h above and substitute it into the cost equation.