A cylinder shaped can needs to be constructed to hold 200 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost.
200 = pi r^2 h
cost= c = 2 pi r h (.03) + (2)pi r^2(.05)
c = 2 pi (.03 rh +.05r^2)
but
h = 200/pi r^2
so
c = 2 pi (.03 r)(200/pir^2) + 2 pi(.05 r^2)
c = 2 pi (6/pir + .05 r^2)
c = 12/r + .1 pi r^2
dc/dr = 0 at min
0 = -12/r^2 + .2 pi r
12 = .2 pi r^3
r = 2.67 cm
check my arithmetic !
To minimize the production cost, we need to find the dimensions of the can that minimize the total surface area.
Let's say the radius of the top and bottom of the can is 'r' centimeters and the height of the can is 'h' centimeters.
The surface area of the sides of the can can be calculated as follows:
Sides Surface Area = 2πrh
The surface area of the top and bottom of the can can be calculated as follows:
Top and Bottom Surface Area = 2πr^2
The total surface area of the can is the sum of the sides and top and bottom surface area:
Total Surface Area = Sides Surface Area + Top and Bottom Surface Area
Total Surface Area = 2πrh + 2πr^2
To find the dimensions that minimize the production cost, we also need to consider the cost of the materials.
The cost of the sides of the can is given as 0.03 cents per square centimeter, so the cost for the sides is:
Cost of Sides = 0.03 * (Sides Surface Area)
The cost of the top and bottom of the can is given as 0.05 cents per square centimeter, so the cost for the top and bottom is:
Cost of Top and Bottom = 0.05 * (Top and Bottom Surface Area)
The total production cost is the sum of the cost of the sides and the cost of the top and bottom:
Total Production Cost = Cost of Sides + Cost of Top and Bottom
Total Production Cost = 0.03 * (2πrh) + 0.05 * (2πr^2)
Now, substitute the equation for the sides surface area and the top and bottom surface area into the total production cost equation:
Total Production Cost = 0.03 * (2πrh) + 0.05 * (2πr^2)
Total Production Cost = 0.06πrh + 0.10πr^2
Since we need to find the dimensions that minimize the production cost, we now have a function in terms of two variables: r (radius) and h (height).
To minimize this function, we need to find its critical points by taking the partial derivatives with respect to each variable and setting them equal to zero.
∂(Total Production Cost) / ∂r = 0
∂(Total Production Cost) / ∂h = 0
After finding the critical points, we can evaluate the function at those points to find the minimum production cost.