A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 250 cubic centimeters of soup. The sides and bottom of the container will be made of syrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for the package that will minimize production cost.
Helpful information:
$h:$ height of cylinder, $r$ : radius of cylinder
Volume of a cylinder: $V=\pi r^{2} h$
Area of the sides: $A=2 \pi r h$
Area of the top/bottom: $A=\pi r^{2}$
To minimize the cost of the package:
Radius:
$\mathrm{cm}$
Height:
$\mathrm{cm}$
Minimum cost:
cents
Answer
\(\boxed{\text{Final Answer: The dimensions that will minimize the production cost are a radius of approximately 3.41 cm and a height of approximately 6.83 cm. The minimum cost is approximately 8.38 cents.}}\)
Step 1 ：Given that the volume of the soup the cylinder must hold is fixed at 250 cubic centimeters, we can express the volume of the cylinder in terms of one variable. Let's choose the radius r as this variable, and express the height h in terms of r using the volume formula \(V=\pi r^{2} h\). This gives us \(h = \frac{250}{\pi r^{2}}\).
Step 2 ：Next, we can express the total cost of the materials in terms of r. The cost of the sides is determined by the area of the sides times the cost per square centimeter of the styrofoam, and the cost of the top is determined by the area of the top times the cost per square centimeter of the paper. This gives us the cost function \(cost = 0.08\pi r^{2} + \frac{20.0}{r}\).
Step 3 ：To find the value of r that minimizes the total cost, we take the derivative of the cost function with respect to r, set it equal to zero, and solve for r. This gives us \(cost' = 0.16\pi r - \frac{20.0}{r^{2}}\). Solving this equation gives us three solutions for r, but only one of them is positive and real, which is approximately 3.41 cm.
Step 4 ：Substituting this value of r into the equation for h gives us the height of the cylinder that minimizes the cost, which is approximately 6.83 cm.
Step 5 ：Substituting this value of r into the cost function gives us the minimum cost, which is approximately 8.38 cents.
Step 6 ：\(\boxed{\text{Final Answer: The dimensions that will minimize the production cost are a radius of approximately 3.41 cm and a height of approximately 6.83 cm. The minimum cost is approximately 8.38 cents.}}\)
