Solving Maximisation Problem with Sections a, b, c

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In summary, the conversation discusses a problem involving finding the derivative of intensity with respect to a variable, given equations involving Pythagorean theorem and cosine. The conversation also clarifies the use of the chain rule and the treatment of variables as constants or parameters in differentiation.
  • #1
tomc612
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Hi,
Ive got a problem I need some help with

View attachment 6076

Ive got sections a, b, c
a) d = \sqrt{{x}^{2}+{R}^{2}}
b) cos\theta = x/\sqrt{{x}^{2}+{R}^{2}}
c) I = 1/{x}^{2}+{R}^{2}
d) x>0

My question is how do i differentiate for I from answer in c) and find 0 and then maximise from there
 

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  • #2
Hello and welcome to MHB, tomc612! (Wave)

a) By Pythagoras, we find:

\(\displaystyle d=\sqrt{x^2+r^2}\quad\checkmark\)

b) Using the definition of cosine as adjacent over hypotenuse, we find:

\(\displaystyle \cos(\theta)=\frac{x}{d}=\frac{x}{\sqrt{x^2+r^2}}\quad\checkmark\)

c) Using the given information regarding intensity $I$, we find:

\(\displaystyle I=k\frac{\cos(\theta)}{d^2}=k\frac{\dfrac{x}{\sqrt{x^2+r^2}}}{x^2+r^2}=k\frac{x}{(x^2+r^2)^{\frac{3}{2}}}\)

where $0<k$ is a constant of proportionality.

d) We must have $0\le x$.

e) Can you now use the quotient and chain rules to find \(\displaystyle \d{I}{x}\)?
 
  • #3
Hi Mark,
thanks for the help,
I see where I went wrong with the proportions as I only includes I= 1/d rather than I= 1/d. Cos\theta

For the differentiation of DI/Dx do we leave the R^2 in the equation or differentiate that in another equation as part of the chain rule.. as in DI/Dx = DR/DX x DI/DR

Thanks
 
  • #4
We can treat $r$ as a constant, or a parameter of the problem. $x$ can vary as the light is raised or lowered, but the radius of the table will remain the same. :D
 

FAQ: Solving Maximisation Problem with Sections a, b, c

How do you determine the sections a, b, and c in a maximization problem?

The sections a, b, and c in a maximization problem are determined by breaking the problem down into smaller parts and identifying the relevant factors and constraints. These sections represent different aspects of the problem that need to be optimized in order to achieve the maximum outcome.

What is the objective function in a maximization problem?

The objective function in a maximization problem is the mathematical expression that represents the goal or desired outcome. It is the function that is being optimized in order to achieve the highest possible value.

How can you solve a maximization problem with sections a, b, and c?

To solve a maximization problem with sections a, b, and c, you can use various mathematical techniques such as linear programming, calculus, or algebra. It is important to identify the relevant variables and constraints for each section and then use appropriate methods to optimize the objective function.

What are some common challenges when solving a maximization problem with sections a, b, and c?

Some common challenges when solving a maximization problem with sections a, b, and c include identifying the correct objective function, determining the appropriate constraints, and selecting the most effective method for optimization. It is also important to consider any real-world limitations or uncertainties that may affect the outcome.

Can you provide an example of a real-world problem that can be solved using maximization with sections a, b, and c?

One example of a real-world problem that can be solved using maximization with sections a, b, and c is resource allocation in a company. The sections a, b, and c could represent different departments or resources within the company, and the objective function could be to maximize profits or minimize costs. By optimizing the allocation of resources across different sections, the company can achieve the best overall outcome.

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