A cylinder shaped can needs to be constructed to hold 1000 cubic centimeters of soup. The material for the sides
of the can costs 0.1 cents per square centimeter. The material for the top is.2 and the bottom is .3 cents per square centimeter. Find the dimensions for the can that will minimize
production cost.
volume = pi r^2 h =1000
so h = 1000/(pi r^2)
side area = 2 pi r h
top area = pi r^2
bottom area = pi r^2
cost = c=0.1*2 pi r h +(.2+.3)pi r^2
or
c = 0.2 pi r h + .5 pi r^2
c = .2 (1000/r) + .5 pi r^2
dc/dr = 0 at min
= -200/r^2 + pi r
r^3 = 200/pi
r = 4 cm
h =1000/(pi r^2) = 1000/(16 pi) = 20
To find the dimensions that will minimize the production cost, we need to consider the cost of the material used to make the can. The cost is directly related to the surface area of the can.
Let's start by determining the formula for the surface area of the can.
A can consists of three parts: the top, the bottom, and the sides. The top and bottom are both circles, and the sides are a rectangle that has been rolled into a cylinder.
The surface area of the top and bottom circles can be calculated using the formula for the area of a circle:
Area = π * r^2
Where "r" is the radius of the circle. Since both the top and bottom circles have the same radius, we will call it "r".
So, the total surface area of the top and bottom can be expressed as:
2 * (π * r^2)
The surface area of the side rectangle can be calculated by multiplying the height of the cylinder by the circumference of the base circle.
The height of the cylinder can be determined by dividing the volume of the can by the area of the base circle.
Given that the volume of the can is 1000 cubic centimeters, to calculate the height, we divide 1000 by the area of the base circle.
The area of the base circle is given by:
Area = π * r^2
So, the height of the cylinder is:
height = 1000 / (π * r^2)
Now, let's calculate the surface area of the sides.
The circumference of the base circle is given by:
Circumference = 2 * π * r
Multiplying the circumference by the height, we get the surface area of the sides:
2 * π * r * height
Now, we can express the total surface area of the can as:
Total Surface Area = 2 * (π * r^2) + 2 * π * r * height
To minimize the production cost, we need to minimize the total surface area. We can treat the cost as a function of the variables r (radius) and height (h), and find the values of r and h that minimize that function.
The cost function can be calculated by multiplying the surface area by the cost per square centimeter for each section of the can:
Cost = (2 * (π * r^2) * 0.1) + (2 * π * r * height * 0.1) + (π * r^2 * 0.2) + (π * r^2 * 0.3)
Simplifying the equation, we get:
Cost = (0.2 + 0.3) * (π * r^2) + (2 * 0.1 * π * r * height)
We need to find the values of r and h that minimize this cost function.
To find the minimum, we can take the derivative of the cost function with respect to each variable (r and h), set the derivatives equal to zero, and solve for r and h.
After finding the values of r and h, substitute them back into the total surface area equation to find the dimensions (radius and height) that minimize the production cost.