Question

You will receive a thumbs up. Be sure to explain steps. 3.) A right circular cylinder can is to be designed to hold 17 cubic inches of a liquid. The cost for the material for the top and bottom of the can is $12 per square inch while the cost for the material of the side of the can is $4 per square inch. Let r=3 represent the radius and h the height of the cylinder. The surface area of the side of a cylinder is A = 2pirh, the surface area of a circle is A =pir^2, and the volume of a cylinder is V =pir^2h. Write the primary and constraint equations needed to minimize the total cost of making the can.

Answer 1

The primary equation for cost is C = 2 * $12 * πr^2 + $4 * 2πrh. The constraint equation for volume is V = πr^2h = 17. These equations help minimize the can's cost while maintaining the required volume.

Step-by-step explanation:

To minimize the total cost of making a can with a fixed volume, we need to establish the primary equation for the cost and the constraint equation for the volume. The primary equation will represent the total cost of making the can based on the surface area of its top, bottom, and side. Given that the cost for material for the top and bottom is $12 per square inch and for the side is $4 per square inch, and given the formulas for the area of a circle (A = πr^2) and the side of a cylinder (A = 2πrh), the total cost C is C = 2 * $12 * πr^2 + $4 * 2πrh.

The constraint equation for volume V given that the cylinder must hold 17 cubic inches is V = πr^2h = 17.

This system of equations can then be used to find the values of r and h that minimize C while satisfying the volume constraint.