# Find the dimensions that will minimize the cost of the material?

##
A cylindrical container that has a capacity of #10\text( m)^3# is to be produced.

- The
*top and bottom of the container* are to be made of a material that *costs *#$20# per square meter,
- while the
*side of that container* is to be made of a material *costing *#$15# per square meter.

Find the dimensions that will minimize the cost of the material.

A cylindrical container that has a capacity of

- The
*top and bottom of the container*are to be made of a material that*costs*#$20# per square meter, - while the
*side of that container*is to be made of a material*costing*#$15# per square meter.

Find the dimensions that will minimize the cost of the material.

For students who have not yet learned calculus of two variables, here is the single variable solution.

A (right circular) cylinder has two variables,

Finally, the question asks for the dimensions, so we have

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The volume is given by

Now, the problem can be stated as

Using lagrange multipliers it reads

with stationary points given by

#{(2 c_2 pi r + pi r^2 lambda= 0), (2 c_2 h pi+ 4 c_1 pi r + 2 h pi r lambda = 0), (h pi r^2 - V_0 =0):}#

We know that is a minimum because

and for the found solution has the value

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To find the dimensions that minimize the cost of material, you typically need a specific context or problem statement. The general approach involves setting up a cost function based on the dimensions of the object or structure in question, then using calculus techniques such as differentiation to find the critical points where the cost function reaches a minimum. These critical points can then be analyzed to determine the optimal dimensions. If you provide the specific context or problem statement, I can assist you further in solving it.

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To find the dimensions that will minimize the cost of the material, you need to provide the specific context or details of the problem, such as the shape of the object being constructed, the cost function for the material, and any constraints or limitations. Once those details are provided, mathematical techniques such as calculus optimization can be applied to determine the dimensions that minimize the cost.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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