# Optimization Problem with Cylinder

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In summary, to find the minimum height of the cylindrical tin can with a volume of 16\pi cubic inches, we must first determine the surface area as a function of height only. This can be done by setting the volume equation V=\pir2h equal to 16\pi and solving for r in terms of h. Then, using this value of r, we can find an expression for the surface area of the can as a function of height. To find the minimum value of this function, we can use techniques such as taking derivatives or setting the derivative equal to zero and solving for h. This will give us the minimum height needed to construct the can with the minimum amount of tin.
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## Homework Statement

The volume of a cylindrical tin can with a top and a bottom is to be 16$$\pi$$ cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

## Homework Equations

V=$$\pi$$r2h

## The Attempt at a Solution

So I first tried solving for r in termsof h, but from there I am not sure how I am suppose to do this problem. Could someone please help me?

You are trying to minimize the surface area of the can. What is the surface area of a cylinder of radius r and height h?

You are told that no matter what h and r are, you must always have $$V=16 \pi$$, hence the radius r is dependent on what the height h is. For a given h, what should the radius of the can be?

You should then be able to come up with an expression for the surface area of the can as a function of height only. How can you find the minimum value of a function?

## 1. What is an optimization problem with a cylinder?

An optimization problem with a cylinder involves finding the maximum or minimum value of a certain quantity, such as volume or surface area, while considering the constraints and parameters of a cylinder.

## 2. What are the common constraints in an optimization problem with a cylinder?

The common constraints in an optimization problem with a cylinder include the radius and height of the cylinder, as well as any other given limitations, such as available material or cost.

## 3. What is the formula for the volume of a cylinder?

The formula for the volume of a cylinder is V = πr2h, where r is the radius and h is the height of the cylinder. This formula is commonly used in optimization problems with cylinders.

## 4. How do you solve an optimization problem with a cylinder?

To solve an optimization problem with a cylinder, you can use calculus and set up an equation to represent the quantity to be optimized, then take the derivative and set it equal to zero to find the critical points. These points can then be evaluated to determine the maximum or minimum value.

## 5. What are some real-life applications of optimization problems with cylinders?

Optimization problems with cylinders have many real-life applications, such as finding the optimal dimensions for a water tank or oil pipeline, designing efficient storage containers, and maximizing the volume of a can or bottle while minimizing material costs.

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