A right cicular cylindrical can is to be constructed to have a volume of 57.749 cubic inches (one quart). The sides of the can are to be formed by rolling and welding a strip of metal, which may be purchased in rolls with width equal to the desired height of the can. The material for the sides costs 20 cents per square foot. The welding cost is 1.1 cents per inch. Top and bottom of the can are circles cut from hexagons to minimize the waste. the width of the hexagon is the diameter of the can plus 0.4 inches. the extra inches is crimpled over the sides to form the seal. Crimping costs are 1.6 cents per inch and the material for the ends sells for 30 cents per square foot. The metal for the ends of the cans may be purchased in a rolls which allows for exactly 4 hexagons and exactly 3 hexagons in alternate strips. Find the dimensions of the most economical can which can be constructed to meet these specifications.

I answered this elsewhere. You must have posted it twice, under different names.

To find the dimensions of the most economical can, we need to consider the cost of the material and the welding, as well as the volume of the can. Let's break down the problem into smaller steps:

Step 1: Find the dimensions of the can.
Let's assume the height of the cylinder is 'h' inches, and the radius of the cylinder is 'r' inches. The volume of a cylinder can be calculated using the formula V = πr^2h. In this case, the volume of the can is given as 57.749 cubic inches.

Step 2: Calculate the material cost.
The sides of the can are formed by rolling and welding a strip of metal. The width of the strip is equal to the height of the can, which is 'h'. The cost of the material is given as 20 cents per square foot. First, we need to convert the units. Since we are dealing with cubic inches, 1 square foot is equal to (12^2) * 1 = 144 square inches. So the cost of the material per square inch is 20/144 cents.

Step 3: Calculate the welding cost.
The welding cost is given as 1.1 cents per inch. So the total welding cost depends on the circumference of the can, which is 2πr.

Step 4: Calculate the cost of the top and bottom ends.
The top and bottom of the can are circles cut from hexagons. The width of the hexagon is the diameter of the can plus 0.4 inches, which is (2r) + 0.4. The extra inches are crimped over the sides to form the seal, and the crimping cost is given as 1.6 cents per inch. The material cost for the ends is given as 30 cents per square foot, which we'll need to convert to square inches.

Step 5: Optimize the cost.
To find the dimensions of the most economical can, we need to minimize the total cost, which includes the material cost, welding cost, and cost of the top and bottom ends.

Now, using the above information, you can set up equations using the given costs and dimensions, and then solve for the height 'h' and the radius 'r' that minimize the total cost.