A cylinder shaped can needs to be constructed to hold 200 cubic centimeters of soup. The material for the sides
of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be
thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize
production cost.
Radius of the can:
Height of the can:
Minimum cost:
πr^2h = 200, so h = 200/πr^2
The cost is thus
c = .02*2πrh + .06*2πr^2
= .04πr(200/πr^2) + .12πr^2
= 8/r + .12πr^2
dc/dr = -8/r^2 + .24πr
dc/dr=0 when r ≈ 2.2
Now just crank out h and c
To find the dimensions of the cylinder that will minimize production cost, we need to consider the cost of the material for the sides and the cost of the material for the top and bottom separately.
Let's start by finding the formula for the cost of the material for the sides of the can. The area of the sides of the cylinder is given by the formula:
A_sides = 2πrh
where r is the radius of the cylinder, and h is the height of the cylinder.
The cost of the material for the sides is then given by:
Cost_sides = A_sides * 0.02
Next, let's find the formula for the cost of the material for the top and bottom of the can. The area of each circular face of the cylinder is given by the formula:
A_face = πr^2
The cost of the material for the top and bottom is then given by:
Cost_face = 2A_face * 0.06
To minimize the production cost, we want to minimize the combined cost of the sides and the top/bottom. So, the total cost is given by:
Total_cost = Cost_sides + Cost_face
Now, we need to find the minimum cost. To do that, we can express the total cost in terms of one variable, say the radius r. We'll also use the volume of the cylinder to relate r and h.
The volume V of the cylinder is given by:
V = A_face * h
= πr^2 * h
Since we want the can to hold 200 cubic centimeters of soup, we have:
πr^2 * h = 200
h = 200 / (πr^2)
Substituting this expression for h into the formula for the cost of the sides (A_sides) and the total cost (Total_cost), we get:
A_sides = 2πr(200 / (πr^2))
= 400 / r
Total_cost = (400 / r) * 0.02 + 2πr^2 * 0.06
To find the minimum cost, we can take the derivative of Total_cost with respect to r, set it equal to 0, and solve for r.
d(Total_cost) / dr = -0.04/r^2 + 0.12πr = 0
Multiplying through by r^2:
-0.04 + 0.12πr^3 = 0
Simplifying:
r^3 = 0.04 / (0.12π)
r = (0.04 / (0.12π))^(1/3)
Now we can substitute the value of r into the expression for h to find the height h of the can:
h = 200 / (πr^2)
Finally, we can calculate the minimum cost by substituting the values of r and h into the formula for Total_cost:
Minimum_cost = (400 / r) * 0.02 + 2πr^2 * 0.06
Now you can calculate the values for the radius of the can, the height of the can, and the minimum cost.