Optimization- oh how the brain hurts

In summary, the problem is to find the radius of a cylinder that produces the minimum surface area when two hemispheres are attached to its ends. The total volume of the solid is 12 cm^3 and the equations for the volume of a sphere and cylinder are given. The solution involves setting the volume equations equal to each other and solving for the height, then using that value to find the surface area.
  • #1
griffon
3
0
Q: A solid if formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cm^3. Find the radius of the cylinder that produces the minimum surface area.

OK, I got about halfway through my problem before I got lost.
V of shpere= 4/3(pi)r^3
V of cylinder= (pi)r^2(h)
total V= 2(4/3*pi*r^30) + (pi*r^2*h)

12=pi*r^2(2*4/3*r+h) which turns into 12/pi*r^2=8/3*r+h
so h=(12/pi*r^2) - (8/3*r)

Now I'm lost.
 
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  • #2
And what is the surface area as a function of r and h?
 
  • #3
The problem was to minimize the surface area. You have done nothing with the surface area. What is the surface area of a sphere of radius r? What is the area of the curved surface of a cylinder or radius r and length h?
 

What is optimization?

Optimization is the process of finding the best possible solution to a problem or maximizing a desired outcome. It involves evaluating different options and selecting the one that yields the most favorable result.

How does the brain play a role in optimization?

The brain is responsible for processing information and making decisions. In optimization, the brain uses cognitive processes, such as problem-solving and critical thinking, to analyze and evaluate the available options and make a decision that will lead to the best outcome.

What factors can affect optimization?

There are several factors that can influence optimization, such as the complexity of the problem, the amount of available information, and the individual's cognitive abilities and biases. External factors, such as time constraints and external pressure, can also impact the optimization process.

How can we improve our optimization skills?

Practice and experience can enhance our optimization skills. Engaging in activities that require problem-solving and decision-making can help train our brains to become better at optimization. Additionally, being aware of our cognitive biases and taking the time to thoroughly analyze and evaluate options can also improve optimization abilities.

Can optimization be applied to different areas of life?

Yes, optimization can be applied to various aspects of life, including personal and professional decision-making, problem-solving, time management, and resource allocation. It is a valuable skill that can help individuals achieve their goals and make the most efficient use of their resources.

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