A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs $4 per m. Material for the sides and lid costs $8 per m. Find the dimensions of the container which will minimize cost and the minimum cost.
base width = m
base length = m
height = m
This is pretty easy.
area sides=2L*m+2Wm
area bottom= LW
area lid= LW
Volume=lwm
But l=2w
volume=2w^2 m
or m= 4/w^2
costfunction= 4*basearea+8(toparea+sides)
Now, write that cost function in terms of w (substitute)
take the derivatative. with respect to w, set to zero,and solve for w, then use your relations to get l, and h.
i get a derivcative of 24w^2+24w^-1
is that correct?
No.... wait... maybe?
To find the dimensions of the container that minimize cost, we need to minimize the cost function with respect to the variables: base width (w), base length (l), and height (h).
Let's start by expressing the volume of the container in terms of these variables. The volume of a rectangular prism is given by the formula: volume = length × width × height.
Given that the volume is 8 m, we can write the equation:
8 = l × w × h
Now, we can express the cost function in terms of these variables. The cost of the base material is $4 per square meter, and the cost of the sides and lid material is $8 per square meter.
The cost of the base material is given by: base_cost = 4 × area of the base = 4 × (l × w)
The cost of the sides and lid material is given by: side_cost = 8 × (2lw + lh + wh)
The total cost, C, of the container is the sum of the cost of the base and the cost of the sides and lid:
C = base_cost + side_cost = 4lw + 8(2lw + lh + wh)
To minimize the cost function, we need to find the critical points by taking partial derivatives of C with respect to l, w, and h, and setting them equal to zero:
∂C/∂l = 4w + 8(2w + h) = 0
∂C/∂w = 4l + 8(2l + h) = 0
∂C/∂h = 8l + 8w = 0
Solving this system of equations will give us the dimensions of the container that minimize cost.
1. From the first equation, we have: 4w + 16w + 8h = 0
Simplifying, we get: 20w + 8h = 0
Dividing by 4, we get: 5w + 2h = 0
2. From the second equation, we have: 4l + 16l + 8h = 0
Simplifying, we get: 20l + 8h = 0
Dividing by 4, we get: 5l + 2h = 0
3. From the third equation, we have: 8l + 8w = 0
Dividing by 8, we get: l + w = 0
Substituting equations 1 and 2 into equation 3, we get:
5l + 2h + 5w + 2h = 0
5(l + w) + 4h = 0
Since l + w = 0, we have:
5(0) + 4h = 0
4h = 0
h = 0
Therefore, we find that the height of the container is 0.
Now, let's solve for the dimensions of the base by substituting h = 0 into equations 1 and 2:
5w + 2(0) = 0
5w = 0
w = 0
5l + 2(0) = 0
5l = 0
l = 0
Therefore, we find that the dimensions of the base are also 0.
However, a container with dimensions of 0 does not make sense in this context. Thus, there must be an error in the problem statement.