A manufacturer needs to produce a cylindrical can with a volume (capacity) of 1000cm cubed. The top and the bottom of the container are made of material that costs $0.05 per square cm, while the side of the container is made of material costing $0.03 per square centimeter. Find the dimensions of the container that will minimize the company's cost of producing this container. What is the minimum cost?
done, see your previous post of this
btw, I notice you had to find the dimensions as well as the minimum cost
but once you have r, just go back into h = ...
To find the dimensions of the container that will minimize the cost, we can start by expressing the cost function in terms of the dimensions of the container. Let's assume that the height of the cylinder is "h" and the radius of the cylinder is "r".
The cost of the top and the bottom of the container is proportional to their surface area, which is the area of a circle with radius "r". So the cost for the top and bottom can be expressed as:
Cost_top_bottom = 2 * (0.05 * π * r^2) = 0.1 * π * r^2
Similarly, the cost of the side of the container is proportional to its cylindrical surface area, which is the circumference of a circle with radius "r" times the height "h". So the cost for the side can be expressed as:
Cost_side = 0.03 * (2 * π * r * h) = 0.06 * π * r * h
The total cost of producing the container is the sum of the cost for the top and bottom and the cost for the side:
Cost_total = Cost_top_bottom + Cost_side = 0.1 * π * r^2 + 0.06 * π * r * h
Now we want to minimize this cost function with respect to the dimensions "r" and "h". To do that, we can take partial derivatives with respect to "r" and "h" and set them equal to zero.
∂Cost_total/∂r = 0.1 * 2 * π * r + 0.06 * π * h = 0
∂Cost_total/∂h = 0.06 * π * r = 0
From the second equation, we can see that r = 0, which means either the cost is zero (which is not possible) or h is zero (which also doesn't make sense because we need a non-zero height for the cylinder).
Therefore, we can only focus on the first equation to solve for r:
0.1 * 2 * π * r + 0.06 * π * h = 0
0.2 * π * r = -0.06 * π * h
r = (-0.06 * π * h) / (0.2 * π)
r = -0.3 * h
Now let's substitute this expression for r back into the cost function, and simplify it further:
Cost_total = 0.1 * π * r^2 + 0.06 * π * r * h
= 0.1 * π * (-0.3 * h)^2 + 0.06 * π * (-0.3 * h) * h
= 0.09 * π * h^2 - 0.054 * π * h^2
= 0.036 * π * h^2
Now we want to minimize the cost function, so we need to find the critical points by setting the derivative of the cost function with respect to "h" equal to zero:
∂Cost_total/∂h = 0.036 * π * 2 * h = 0
h = 0
Again, we have found that h = 0, which means that the cost is zero (which is not possible) or r is zero (which also doesn't make sense because we need a non-zero radius for the cylinder).
Therefore, in this case, we cannot find a minimum cost because the given constraints do not allow for it.
To summarize, the dimensions that would minimize the company's cost of producing this container cannot be determined under the stated constraints, and hence, the minimum cost cannot be calculated.