Question

A rectangular storage container with an open top is to have a volume of 26 cubic meters. The length of its base is twice the width. Material for the base costs 12 dollars per square meter. Material for the sides costs 9 dollars per square meter. Find the cost of materials for the cheapest such container

Answer 1

Can you write the equation for the cost function?

Answer 2

l = length
h = height
w = width
v = volume

Given:
v = 40
l = 2w

Substituting v=40 and l=2w into volume formula:
V = l * w * h
40 = 2w * w * h
40 = 2w^2 * h

Solving for h:
40/2w^2 = h
20/w^2 = h

Determine the cost function:
Cost = 2*(end area)*$6 + 2*(side area)*$6 + (bottom area)*$40
Cost = 12*(h*w) + 12*(h*l) + 40*(w*l)

Substitute l=2w:
Cost = 12hw + 12*(h*2w) + 40*(w*2w)
Cost = 12hw + 24hw + 80w^2
Cost = 36hw + 80w^2

Substitute h=20/w^2:
Cost = 36(20/w^2)w + 80w^2
Cost = 36(20/w) + 80w^2
Cost = 720/w + 80w^2

To find the min/max, take the derivative:
Cost' = -720/w^2 + 160w

and find the zeros:
0 = -720/w^2 + 160w
720/w^2 = 160w
720/160 = w^3
4.5 = w^3
1.65096362 = w

Using l=2w:
l = 2* 1.65096362
l = 3.301927249

Using h=20/w^2:
h = 20/(1.65096362)^2
h = 20/(1.65096362)^2
h = 20/2.7256808745635044
h = 7.337616148

If we take the second derivative:
Cost' = -720w^(-2) + 160w
Cost'' = 1440w^(-3) + 160

At the point w=1.65096362, the function is concave up,
thus this is a local minimum.

To determine the cost at this point:
Cost = 720/w + 80w^2
Cost = 720/(1.65096362) + 80*(1.65096362)^2
Cost = 654.16


Therefore, the solution is:
Width = 1.65096362
Length = 3.301927249
Height = 7.337616148
Cost = 654.16

Answer 3

holy cow... Thank you! This problem has been giving me trouble for almost 45 minutes now

Answer 4

unfortunately , the answer they provided is wrong...compare the variables in your problems to theirs...For example your volume = 26 m^3....they used 40

Answer 5

O I see, ya they used different variables... At least I know how to solve it now haha

Answer 6

medal guys

Answer 7

be nice