Minimizing the cost of a gas cylinder

In summary, to minimize the cost of constructing a gas cylinder with hemispherical ends, we need to have a cylinder with dimensions of radius (r) = (3V/4π)^(1/4) and height (h) = (4V/πr^2)^(1/2). Thank you for your question and I hope this helps.
  • #1
leprofece
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determine as dimensions of a gas cylinder of volume "V" for .an industry, whose coast is minimal, if the same Eastern pair formed a body par topped cylindrical half balls at the ends. It is known that the cost of construction par m2 of these hemispheres is 6 times greater than for the cylindrical body.
 
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  • #2

Thank you for your question. I can provide you with the dimensions of the gas cylinder based on the given information.

Firstly, let's understand the constraints given in the problem. We have a cylindrical body with two hemispherical ends. The cost of constructing the hemispheres is 6 times greater than the cylindrical body. Let's assume the radius of the cylinder to be r and the height to be h.

The volume of the cylinder can be calculated as V = πr^2h. Since the hemispheres are formed at the ends, the total volume of the cylinder will be equal to the sum of the volumes of the two hemispheres and the cylindrical body. This can be represented as V = 2(2/3 πr^3) + πr^2h.

Simplifying this equation, we get V = (4/3 πr^3) + πr^2h. Now, we know that the cost of constructing the hemispheres is 6 times greater than the cylindrical body. This means that the cost of constructing the hemispheres is 6 times the cost of constructing the cylindrical body, which can be represented as 6πr^2h.

Therefore, the total cost of constructing the gas cylinder can be expressed as (4/3 πr^3) + 7πr^2h. To minimize the cost, we need to minimize the surface area of the cylinder. The surface area of the cylinder can be calculated as SA = 2πrh + 2πr^2. We can substitute the value of h from the volume equation and simplify it to get SA = (2V/r) + 2πr^2.

To minimize the surface area, we need to take the derivative of SA with respect to r and equate it to 0. This will give us the value of r which minimizes the surface area. Solving this equation, we get r = (3V/4π)^(1/4).

Using this value of r, we can calculate the height of the cylinder as h = (4V/πr^2)^(1/2). Therefore, the dimensions of the gas cylinder with minimal cost will be:

Radius (r) = (3V/4π)^(1/4)
Height (h) = (4V/πr^2)^(1
 

Related to Minimizing the cost of a gas cylinder

1. How can I minimize the cost of a gas cylinder?

There are a few ways to minimize the cost of a gas cylinder. One way is to shop around and compare prices from different suppliers. You can also consider purchasing in bulk, as this often results in a lower price per cylinder. Additionally, choosing a smaller, more efficient cylinder size can also save on costs in the long run.

2. Is there a difference in cost between different types of gas cylinders?

Yes, the cost of a gas cylinder can vary depending on the type of gas it contains. For example, cylinders containing high purity gases may be more expensive than those containing standard industrial gases. It is important to consider your specific needs and usage when selecting the right type of gas cylinder to minimize costs.

3. Can I negotiate the cost of a gas cylinder with a supplier?

In some cases, it may be possible to negotiate the cost of a gas cylinder with a supplier. However, this may not always be an option and the final cost may depend on factors such as market demand and the supplier's pricing policies. It is always worth asking about potential discounts or promotions that may help to minimize the cost.

4. Are there any hidden costs associated with gas cylinders?

Yes, there may be additional costs associated with gas cylinders, such as delivery fees or rental charges for the cylinder itself. It is important to inquire about these costs when obtaining quotes from suppliers to get a more accurate understanding of the overall cost.

5. Can I reduce the cost of a gas cylinder by recycling or refilling it?

In some cases, recycling or refilling a gas cylinder can be a cost-effective option. However, this may not be possible for all types of cylinders and it is important to ensure that the cylinder is properly inspected and approved for reuse before doing so. It is also important to follow all safety protocols when handling and storing gas cylinders.

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