Calc derivatives - Minimum total surface area in a box of V = 160 ft2

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SUMMARY

This discussion focuses on optimizing the dimensions of an open-top box with a square base to achieve a minimum total surface area while maintaining a volume of 160 cubic feet. The variables involved are the side length of the base (X) and the height (Y) of the box. The surface area is expressed as a function of these variables, and calculus, specifically derivative applications, is used to find the minimum surface area. The key takeaway is that by setting up the surface area equation and applying derivatives, one can determine the optimal dimensions for the box.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly derivatives
  • Familiarity with geometric formulas for surface area of squares and rectangles
  • Knowledge of volume calculations for three-dimensional shapes
  • Ability to formulate equations based on given constraints
NEXT STEPS
  • Study the application of derivatives in optimization problems
  • Learn how to derive surface area formulas for various geometric shapes
  • Explore methods for solving constrained optimization problems
  • Investigate real-world applications of calculus in engineering and design
USEFUL FOR

Students studying calculus, particularly those focusing on optimization problems, as well as educators and tutors looking for examples of applying derivatives to real-world scenarios in geometry.

Dave_1420
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Hi everyone! I'm new to online math forums. I wonder if anyone can give me a hand on this - it would be greatly appreciated.

Thank you in advance!

Dave

Homework Statement



If a box with a square base and an open top is to have a volume of 160 cubic feet, find the dimensions of the box having the minimum total surface area. Use calculus to find the solution.

Homework Equations



If X are the sides of the base, and Y is the height of the box, what would be the formula to find surface? How can the minimum surface be calculated using derivative applications?

The Attempt at a Solution



Surface = minimum possible
 
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In terms of x and y, formulate an expression for the total outer surface area of the box.
 
If "x" is the length of a side of the square base, y is the height of the box, then the base is an "x by x" square and there are four sides that are "x by y" rectangles. Do you know how to find the area of squares and rectangles?
 

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