Calculus - optimization problem

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SUMMARY

The forum discussion revolves around solving an optimization problem for an aquarium's dimensions to minimize costs. The cost function is defined as total cost = 2hw + 2hl + 5wl, where h is height, w is width, and l is length. The volume constraint is given by V = whl. The user struggles to derive a single-variable equation from these two equations and to apply partial derivatives effectively. The discussion highlights the importance of correctly formulating the cost function and understanding the relationships between the variables involved.

PREREQUISITES
  • Understanding of optimization problems in calculus
  • Familiarity with cost functions and constraints
  • Knowledge of partial derivatives
  • Experience with volume equations in three-dimensional geometry
NEXT STEPS
  • Review the method for deriving equations from constraints in optimization problems
  • Study the application of partial derivatives in multi-variable functions
  • Learn about the Lagrange multiplier method for constrained optimization
  • Practice solving similar optimization problems involving cost minimization
USEFUL FOR

This discussion is beneficial for students studying calculus, particularly those focusing on optimization problems, as well as educators and tutors looking for examples of cost minimization in real-world applications.

Pearce_09
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ok, i have an aquarium of volume V... where the cost of the base is 5 times of the sides... let h- height , w- width, l - lenght...
therefore 2hw + 2hl is the cost of the sides and 5wl is the cost of the base
therefore
total cost = 2hw + 2hl + 5wl --> min
and i want find the dimensions to minimize cost.
I know how to do problems like these I've done many, its just this one is somehow bothering me...
i need to solve for 1 variable then i have a formula with 2 variables.. then i take the partial derivatives to see if my dimensions that i find indeed minimize cost. but for some reason i can't even get past my total cost formula... help please
 
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You know the volume formula, so that in conjunction with your cost forumula will provide you with one variable in terms of another and bingo
 
V= whl is one equation. Can you use the "lagrange multiplier" method?
 
i know how to use lagrange.. but i was told not to for this question
 
I use my cost equation to solve for h and got
h = -5wl / 2w + 2l
and when i sub that into volume and partial derive but it doesn't make any sense.. I get l = w and then that implies that h = -10/8 ... not possible to have a negative number
 
No, you don't have a cost equation. You only have a cost function that you want to minimize. You don't know what it is equal to. (It certainly isn't equal to 0!) The only equation you have is V= whl
 
im missing an equation then, which I am having trouble finding...
 
I just don't understand how to find this other equation
 

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