Finding Extreme Volumes of V(x): A Box Model

In summary, the purpose of finding extreme volumes of V(x) using a box model is to accurately determine the maximum and minimum possible values for the volume of a given box. This is achieved by creating a mathematical representation of the box and using calculus and optimization techniques. The extreme volumes can be affected by factors such as the dimensions of the box and any constraints on the volume. The box model can be applied to any shape of box and has various real-world applications in fields such as engineering, physics, chemistry, and economics.
  • #1
Jan Hill
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Homework Statement



The function V(x) = x(14-2x)(15-x) 0<x<7 models the volume of a box
Find the extreme values of V

and tell whether these represent the smallest and largest volumes or only the smallest or only the largest volumes

Homework Equations





The Attempt at a Solution




I found the derivative of the function to be 210 - 88x + 6x^2 and if I set this equal to 0,
I get 0 = 210 -88x + 6x^2
210 = 88x -6x^2

but where do I go from here?
 
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  • #2
solve for x using quadratic formula.
 

FAQ: Finding Extreme Volumes of V(x): A Box Model

1. What is the purpose of finding extreme volumes of V(x) using a box model?

The purpose of finding extreme volumes of V(x) using a box model is to accurately determine the maximum and minimum possible values for the volume of a given box. This can be helpful in various scientific and engineering applications, such as determining the optimal size and shape for a container or packaging design.

2. How is the box model used to find extreme volumes?

The box model involves creating a mathematical representation of a box by defining its dimensions and using variables to represent the length, width, and height. By using calculus and optimization techniques, the extreme values of the volume can be determined by taking the derivative of the volume function and setting it equal to zero.

3. What factors affect the extreme volumes of V(x) in a box model?

The extreme volumes of V(x) in a box model can be affected by various factors, such as the dimensions of the box, the material it is made of, and any constraints on the volume (e.g. the box must have a certain volume or surface area). These factors can impact the optimal dimensions of the box and thus, the extreme volumes.

4. Can the box model be used for any shape of box?

Yes, the box model can be used for any shape of box as long as the dimensions and constraints are accurately defined. It can be applied to rectangular, square, circular, or irregular shaped boxes.

5. What are some real-world applications of finding extreme volumes using a box model?

Finding extreme volumes using a box model has various real-world applications, such as in engineering and architecture for designing optimal packaging, containers, and building structures. It can also be applied in physics and chemistry for determining the maximum and minimum volumes of gases and liquids in containers. Additionally, it can be used in economics for optimizing production and distribution costs for boxed products.

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