Arectangular storage container with an open top is to have a volume of 20 cubic meters. the length of its base is twice the width. material for the base costs 20 dollars per square meter. material for the sides costs 7 dollars per square meter. find the cost of materials for the cheapest such container.

The volume of a rectangular box is calculated by multiplying the length, width and the height of the box. For this case, it is written as:

20 = lwh

From the problem statement, the length of its base is twice the width. Making the expression for volume.

20 = (2w)wh = 2(w^2)(h)
h = 10/(w^2)

Side A: l *h = (2w)(10/(w^2)) = 20/w
Side B: w*h = w*(10/(w^2)) = 10/w

Base = l*w = 2w*w = 2w^2
Total Area of Side A: 2 * (20/w) = 40/w
Total Area of Side B: 2 * (10/w) = 20/w
Total Area = (40/w)+(20/w) + 2w^2 = 60/w + 2w^2
 
Total area weighted cost = 7*(60/w) + 20*(2w^2) = 420/w + 40w^2
 
Find the critical point: 1. Take derivative. 2. Set to zero. 3. Solve for w.
   
Derivative = 80w - (420/(w^2)) = 0
80w = 420/w^2 --> 80w^3 = 420
w^3 = 420/80
w = apprx (1.738)
 
Re-input into cost formula: 420/1.738 + 40*(1.738)^2 = apprx (362.48)

the law of definite proportions states samples of a compound will always contain the same proportion of elements by mass. in other words, a given compound will always contain the same elements in the same proportions by mass.the law of multiple proportions states when two elements combine with each other to form more than one compound, then the weights of one element that combine with a fixed weight of the other are in a ratio of small whole numbers.