Finding the dimensions of a rectangular box

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SUMMARY

The discussion focuses on optimizing the dimensions of a rectangular box with a volume of 144 cubic inches, where the base is twice as wide as it is deep. The box's cost is influenced by the material used, with the bottom being three times more expensive than the sides and top. Participants emphasize the need for two equations: one for volume and another for cost, which is derived from the surface area of the box. The key to solving the problem lies in establishing these equations and applying optimization techniques.

PREREQUISITES
  • Understanding of volume calculations for rectangular prisms
  • Knowledge of surface area formulas for boxes
  • Familiarity with optimization techniques in calculus
  • Ability to set up and solve systems of equations
NEXT STEPS
  • Study the formula for the volume of a rectangular box: V = length × width × height
  • Learn how to calculate the surface area of a rectangular box: SA = 2lw + 2lh + 2wh
  • Explore optimization methods, particularly using derivatives to find minimum cost
  • Investigate real-world applications of cost minimization in manufacturing
USEFUL FOR

Students in mathematics or engineering, particularly those studying optimization problems, as well as professionals involved in manufacturing and cost analysis.

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I'm worried about the process of solving this problem, can anyone help me?

We are to make a rectangular box, including the top, that has a volume of 144 cubic inches and for which the base is twice as wide as it is deep. The bottom, which must be strong, is made of a material that is three times as expensive as that used for the sides and the top. Find the dimensions of the box that minimize its cost.

I know I must have two equations with two variables in total but I'm not sure what to do after that. Maybe 144=(2x)(x)(y)? and another

Thanks to anyone who can shed some light on this for me, I've been trying for a bit but not sure where to start.
 
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You wrote down an equation for the volume of the box. If you want to minimize the cost, you also need an equation for the cost. That will depend on the amount of material you need, in other words the surface area of the box.
 

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