Optimization find radius problem

In summary, to find the radius of the cylinder that produces the minimum surface area, we can use the formula for surface area and the volume of the solid, and set the derivative of the surface area to zero to find the critical point. Solving for r gives us a value of approximately 1.2, which can be tested to confirm that it is the minimum.
  • #1
calvinnn
9
0
optimization problem!

OKOK running out of time! CAn anyone please help me with this problem:

Surface Area A solid os formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radiusof the cylinder that produces the minimum surface area.

if anyone can help me with this, ill be VERY grateful!
 
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  • #2
Exactly what have you tried so far?
 
  • #3
1. Call the length of the cylinder l and the radius r. Write down the formula for the volume of the figure (its a cylinder and a sphere) and set that equal to 12. Solve for l.

2. Write down the formula for surface area (again, lateral area of a cylinder, area of a sphere) and replace l by the formula from 1 so that you have a function of r only.

3. Find the value of r that minimizes that function.
 
  • #4
Q: Surface Area A solid os formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radiusof the cylinder that produces the minimum surface area.

A:

SA=2 π r2 + 2 π r h

V=πr^2h, V=12

h=12/(πr^2)

thus, SA=(2πr^2)+(2πr)(12/(πr^2))

SA'=4πr-(24/r^2) To find the minimum, set SA' to zero.

0=4πr-(24/r^2) r=6^(1/3)/π^(1/3), approx. 1.2

Test a point on either side of r=1.2 to make sure that it is a minimum.
 
Last edited:

FAQ: Optimization find radius problem

What is the optimization find radius problem?

The optimization find radius problem is a mathematical problem that involves finding the optimal radius of a circle that maximizes or minimizes a given objective function. This problem is commonly encountered in geometry, physics, and engineering.

How is the optimization find radius problem solved?

The optimization find radius problem is typically solved using calculus and optimization techniques. The first step is to set up the objective function, which represents the quantity that needs to be maximized or minimized. Then, the derivative of the objective function is taken and set equal to zero to find the critical points. Finally, the critical points are evaluated to determine the optimal radius.

What are some real-world applications of the optimization find radius problem?

The optimization find radius problem has many practical applications, such as finding the optimal size of a water tank, determining the ideal length of a bridge suspension cable, and maximizing the area of a garden. It is also used in fields such as economics, where it can be used to find the optimal price of a product.

What are the limitations of the optimization find radius problem?

One limitation of the optimization find radius problem is that it assumes a perfect circle, which may not always be the case in real-world scenarios. It also assumes that the objective function is continuous and differentiable, which may not always be true. Additionally, the problem may have multiple solutions or no solution at all, making it challenging to find the optimal radius.

Can the optimization find radius problem be solved using other methods besides calculus?

Yes, the optimization find radius problem can also be solved using other methods such as linear programming, genetic algorithms, and numerical methods. However, these methods may not always provide an exact solution and may require additional assumptions or approximations.

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