# Relationships between variables, for use of optimization

• Monkey618
In summary, the relationship between the height and radius of a steel drum that will minimize the surface area is r = h/2. This is found by taking the derivative of the surface area equation with respect to radius, setting it equal to 0, and solving for r.
Monkey618

## Homework Statement

Build a steel drum (right circular cylinder) of fixed volume V. The cost of the material is the same for the top and sides (same gage and same cost) and disregard waste. What is the relationship between the height of the radius that will minimize the surface area of the cylinder?

## Homework Equations

Volume of cylinder = πr²h
Surface area of cylinder = 2πr² + 2πrh

## The Attempt at a Solution

Find the volume, because you have no actual value it is simply V:
V = πr²h

Set in terms of one variable:
h = V / (πr²)

Find surface are with respect to the volume of the cylinder:
S = 2πr² + 2πrh

therefore, in terms of a single variable r:
S = 2πr² + 2πr (V / πr²)

Take the derivative with respect to r so that you can find where surface area has a minimum.
S = 2πr² + ((2πrV) / (πr²))
dS = 4πr + ( πr²(2Vπ) - 2Vπr(2πr) ) / (πr²)² * dh/dr

Set the derivative to 0 and solve. But because I have no actual values, I solved for dh/dr. Was this correct?

0 = 4πr + ( πr²(2Vπ) - 2Vπr(2πr) ) / (πr²)² * dh/dr

-4πr = + ( πr²(2Vπ) - 2Vπr(2πr) ) / (πr²)² * dh/dr

-4πr / [( πr²(2Vπ) - 2Vπr(2πr) ) / (πr²)²] = dh/dr

Simplify

dh/dr = -4πr / [( πr²(2Vπ) - 2Vπr(2πr) ) / (πr²)²]

dh/dr = -4πr / [(-2V)/(r²)]

dh/dr = (2πr³) / V

Is this a proper relation between height and radius for the minimal surface area of the afore mentioned steel drum? I tend to mess up my derivatives and such in class.

You are aware that
$$\frac{2\pi rV}{\pi r^2}= \frac{2V}{r}$$
aren't you? That's much easier to work with.

Ah. I am aware of it, yes. I did not notice notice it before taking the derivative. It would have made my life easier.

With the way that I've already done it, am I correct?

EDIT:

Okay, I reworked the problem and got this. I’m pretty sure that this is correct.

V = πr²h
S = 2 πr² + 2 πrh

h = V/(πr²)

so ::
S = 2 πr² + 2 πr(V/πr²)

S = 2 πr² + (2Vπr)/(πr²)
S = 2 πr² + (2V)/r

taking the derivative, I get:

ds/dr = 4 πr - 2V/r²

set to 0 and solve

0 = 4 πr - 2V/r²

2V/r² = 4 πr

r³ = (V) / (πr)

substitute in what V actually is:

r³ = (πr²h) / 2π

r³ = (r²h) / 2

r³/r² = (h) / 2

r = h/2

Last edited:

## 1. What is the relationship between two variables?

The relationship between two variables refers to how changes in one variable affect changes in the other variable. This relationship can be positive, meaning that both variables increase or decrease together, or negative, meaning that as one variable increases, the other decreases.

## 2. How can I determine the strength of the relationship between two variables?

The strength of the relationship between two variables can be determined by calculating the correlation coefficient, which measures the degree of linear relationship between the two variables. A correlation coefficient of 1 indicates a perfect positive relationship, while a coefficient of -1 indicates a perfect negative relationship. A coefficient of 0 indicates no relationship between the variables.

## 3. How can I optimize the relationship between two variables?

To optimize the relationship between two variables, you can use various techniques such as regression analysis, which helps to identify the best-fit line that represents the relationship between the variables. This can also involve manipulating the values of one variable to see how it affects the other variable and finding the optimal values that result in the desired outcome.

## 4. Can relationships between variables change over time?

Yes, relationships between variables can change over time. This can happen due to various factors such as external influences, changes in the environment, or changes in the variables themselves. It is important to continually monitor and analyze the relationship between variables to adapt to any changes and optimize for the desired outcome.

## 5. Are there any limitations to using optimization techniques for relationships between variables?

Yes, there are limitations to using optimization techniques for relationships between variables. These techniques assume a linear relationship between the variables, which may not always be the case. Additionally, other variables or factors may influence the relationship, making it difficult to optimize solely based on the relationship between two variables. It is important to consider all factors and use multiple techniques for optimization to get the best results.

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