Expression of cost function.
A rectangular box, without top, has the volume of 10 m^3. The length of it's base is twice it's width. Material for base costs 10$/m^2 and material for sides costs 6$/m^2. Express the cost of materials as a function of the base's width.
The volume (v)= Length*width*height = L*w*h= 10
We know that L=2w
Cost for base Cb=$10/m^2
Cost fro sides Cs=$ 6/m^2
Then the cost C= cost of base + cost of all sides
Then C = Cb * area of the base + Cs area of all 4 sides
Area of base= L*w= 2w*w= 2w^2
Area of 2 sides= 2 h*w
Area of the other 2 sides= 2h*L = 2h*2w= 4hw
But we know that V=L*w*h=2w^2*h=10
==> h= 10/2w^2= 5/x^2
Then Area of all sides= 2w(5/w^2)+4w(5/x^2)=30/w
Then the cost= 10(2w^2)+6(30/w)
First, let's establish the followings:
-the width of the base is w;
-the length of the base is 2w;
-the height of the box is h.
We'll calculate the costs for the sides and the base.
To calculate the cost of the sides, first let's calculate the areas of the sides.
Two sides will have the area: w*h.
The other 2 sides will have the area: 2w*h
The cost of the material of the sides will be:
C1 = 6 (2*w*h + 2*2w*h)
The area of the base is: 2w*w=2w^2
The cost of the base is: C2=10*2w^2
The total cost is:
C = C1+C2
C = 6 (2*w*h + 2*2w*h) + 10*2w^2
We'll open the brackets and we'll have:
C = 12w*h + 24w*h + 20w^2
C = 20w^2 + 36w*h
But we have to express the cost as a function of the width of the base, only.
We'll calculate the height h, knowing the value of the volume of the box, which is givn by enunciation.
V = w*2w*h
10 = 2h*w^2
We'll divide by 2 and we'll get:
5 = h*w^2
h = 5/w^2
We'll substitute h into the expression of the cost.
C = 20w^2 + 36w*h, where h = 5/w^2
C = 20w^2 + 36w*5/w^2
C = 20w^2 + 180/w
So, the function of cost, depending on w, is:
C(w) = 20w^2 + 180/w
Let the base width be x. Then its length as given is 2x.
So its volume is = length*width* height = 2x^2*h = 10 m^3. Or
height , h = 10/2x^2.= 5/x^2.
Material cost for base =base area*rate = 2x^2* $10 = 20x^2...........(1)
Material cost for sides = side area * rate = 2(2x+x)(5/x^2)
*$6 = (30/x)6 = 180/x........(2).
Therefore, expressions (1) + espression at(2) givesthe total cost of the material = 20x^2+180/x. = 20(x^2+9/x)