A cylindrical container is to be produced that will have a capacity of 10 metre cubic
. The top and the
bottom of the container are to be made of material costing $2.00 per square meter, while the
side of the container is to be made of material costing $1.50 per square meter. Find the
dimensions that will minimize the total cost of the container.
radius of cylinder ---- r
height of cylinger ---- h
volume = πr^2 h = 10
h = 10/(πr^2)
cost = 2(2πr^2) + 1.5(2πrh) <------ 2 circles plus a rectangle
= 4πr^2 + 3πr(10/(πr^2))
= 4πr^2 + 30/r
d(cost)/dr = 8πr - 30/r^2 = 0 for a min of SA
8πr = 30/r^2
r^3 = 15/(4π)
r = .... , for a minimum cost
now that you have r, go back and find h
To find the dimensions that will minimize the total cost of the container, we need to first determine the mathematical formula that represents the cost of the container.
Let's denote the radius of the top and bottom of the container as "r" and the height of the container as "h".
The formula for the cost of the container can be written as follows:
Cost = Cost of top and bottom + Cost of side
The cost of top and bottom is determined by the area of the circles formed by the top and bottom of the container, while the cost of the side is determined by the curved surface area of the cylindrical part.
1. Cost of top and bottom:
The top and bottom of the container are circles, so the area of each circle can be calculated using the formula:
Area of circle = π * radius^2
The cost of top and bottom can then be calculated as:
Cost of top and bottom = 2 * (Area of circle) * (Cost per square meter)
2. Cost of side:
The side of the container is a rectangle, which can be "unwrapped" into a rectangle with length equal to the circumference of the circles and height equal to the height of the container:
Length of side = 2 * π * r
Height of side = h
The cost of the side can then be calculated as:
Cost of side = (Length of side * Height of side) * (Cost per square meter)
Now, we can create the formula for the total cost of the container:
Total Cost = Cost of top and bottom + Cost of side
= 2 * π * r^2 * (Cost per square meter) + (2 * π * r * h) * (Cost per square meter)
To minimize the total cost, we can take the derivative of the total cost formula with respect to r and h, set them equal to zero, and solve for r and h.
Note: Since the container is specified to have a capacity of 10 meter cubic, we can also use the formula for the volume of a cylinder to find the relationship between r and h:
Volume = π * r^2 * h = 10
We can use this relation to eliminate one of the variables when we take the derivatives.
Solving this system of equations will give us the values of r and h that minimize the total cost of the container.