A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 550 cubic centimeters of soup. The sides and bottom of the container will be made of syrofoam costing 0.03 cents per square centimeter. The top will be made of glued paper, costing 0.05 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 550 cubic centimeters of soup. The sides and bottom of the container will be made of syrofoam costing 0.03 cents per square centimeter. The top will be made of glued paper, costing 0.05 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

Answer 1

To find the dimensions that will minimize the production cost of the cup-of-soup package, we need to first understand the cost of each component (syrofoam and paper) in terms of surface area.

Let's consider the dimensions of the cylinder:
- Let the radius of the circular base of the cylinder be r.
- Let the height of the cylinder be h.

The volume of a cylinder can be calculated using the formula:
Volume = π * r^2 * h

Given that the volume of soup the cup needs to hold is 550 cubic centimeters, we have:
550 = π * r^2 * h

To minimize the cost, we need to minimize the surface area of the cup.

The surface area of the cylinder can be divided into three parts:
1. The circular base (bottom): This will be made of syrofoam.
2. The sides (lateral surface): These will also be made of syrofoam.
3. The circular top: This will be made of glued paper.

The surface area of the bottom and sides can be calculated using the formula:
Surface Area of Bottom and Sides = 2π * r * h + π * r^2

The surface area of the top can be calculated using the formula:
Surface Area of Top = π * r^2

The cost of the syrofoam component (bottom and sides) is 0.03 cents per square centimeter, and the cost of the paper component (top) is 0.05 cents per square centimeter.

Now, we can express the total cost in terms of r and h:
Cost = (Surface Area of Bottom and Sides * 0.03) + (Surface Area of Top * 0.05)

To find the dimensions that minimize the cost, we can use calculus. We need to find the critical points of the cost function by taking partial derivatives with respect to r and h, and setting them equal to zero. Then, we can solve the resulting system of equations to find the values of r and h that minimize the cost.

Once we find the values of r and h that minimize the cost, we can use them to calculate the dimensions of the package.

Answer 2

To minimize the production cost, we need to find the dimensions of the cylindrical cup-of-soup package that will minimize the surface area.

Let's assume the radius of the circular base of the cylinder is "r" and the height is "h".

The volume of the cylindrical cup-of-soup package is given as 550 cubic centimeters, so we have the equation:

Volume = π * r^2 * h = 550

To minimize the production cost, we need to minimize the surface area. The surface area of the cylindrical cup-of-soup package consists of three parts: the curved surface area (the sides), the circular base, and the top.

The curved surface area (the sides) of the cylinder is given by:

Curved Surface Area = 2 * π * r * h

The circular base has an area of:

Base Area = π * r^2

The top area is the same as the base area.

Now, let's write the equation for the total surface area:

Total Surface Area = Curved Surface Area + 2 * Base Area

Total Surface Area = 2 * π * r * h + 2 * π * r^2

Now, let's write the equation for production cost:

Production Cost = Cost of syrofoam * Surface Area of syrofoam + Cost of glued paper * Surface Area of glued paper

Production Cost = 0.03 * (2 * π * r * h + 2 * π * r^2) + 0.05 * (π * r^2)

We need to find the dimensions (r and h) that minimize this production cost.

To continue, we need to find the value of r or h in terms of the other variable using the volume equation. Let's solve the volume equation for r or h.

Using the volume equation:
π * r^2 * h = 550

We can rearrange it to get:
h = 550 / (π * r^2)

Now substitute this value of h into the equation for the production cost:

Production Cost = 0.03 * (2 * π * r * (550 / (π * r^2)) + 2 * π * r^2) + 0.05 * (π * r^2)

Simplifying further:

Production Cost = 0.03 * (1100 / r + 2 * π * r^2) + 0.05 * (π * r^2)

Now, we have the production cost equation in terms of only one variable, r.

To minimize the production cost, we need to differentiate the cost equation with respect to r, set it equal to zero, and solve for r.

d(Production Cost)/dr = 0

Now, differentiate the production cost equation with respect to r:

d(Production Cost)/dr = 0.03 * (-1100 / r^2 + 4 * π * r) + 0.05 * (2 * π * r)

Setting it equal to zero and solving for r:

0.03 * (-1100 / r^2 + 4 * π * r) + 0.05 * (2 * π * r) = 0

Simplifying:

-1100 / r^2 + 4 * π * r = 2 * π * r

Multiplying through by r^2:

-1100 + 4 * π * r^3 = 2 * π * r^3

Bringing all terms to one side:

2 * π * r^3 - 4 * π * r^3 = 1100

-2 * π * r^3 = 1100

Dividing through by -2 * π:

r^3 = -550 / π

Now, take the cube root of both sides to get the value for r:

r = cube root of (-550 / π)

Now that we have the value of r, we can substitute it back into the volume equation to find the value of h:

h = 550 / (π * r^2)

Substituting the value of r we found earlier, we can solve for h.

Finally, we have the values of r and h that will minimize the production cost for the cylindrical cup-of-soup package.

Answer 3

To find the dimensions that will minimize the production cost of the cup-of-soup package, we need to first understand the cost of each component (syrofoam and paper) in terms of surface area.

To minimize the production cost, we need to find the dimensions of the cylindrical cup-of-soup package that will minimize the surface area.

similar question here, just change the numbers