# How do you find the cost of materials for the cheapest such container given a rectangular storage container with an open top is to have a volume of #10m^3# and the length of its base is twice the width, and the base costs $10 per square meter and material for the sides costs $6 per square meter?

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To find the cost of materials for the cheapest such container, we need to minimize the total cost, considering the cost of the base and the cost of the sides.

Given:

- Volume of the container = 10 m³
- Length of the base = twice the width
- Cost of base material = $10 per square meter
- Cost of side material = $6 per square meter

Let's denote:

- Length of the base as ( L )
- Width of the base as ( W )
- Height of the container as ( H )

From the given information, we know that the volume of the container (( V )) is given by: [ V = L \times W \times H ]

Given that ( L = 2W ), we can rewrite the volume equation as: [ V = 2W \times W \times H = 2W^2H ]

Since the volume (( V )) is given as 10 m³, we have: [ 10 = 2W^2H ]

We need to minimize the cost function ( C ), which is the sum of the cost of the base and the cost of the sides: [ C = 10WL + 6(2LH + 2WH) ]

Substituting ( L = 2W ) into the cost function, we get: [ C = 10(2W)W + 6\left(2(2W)H + 2W(H)\right) ] [ C = 20W^2 + 6(4WH + 2WH) ] [ C = 20W^2 + 6(6WH) ] [ C = 20W^2 + 36WH ]

Now, we can express ( H ) in terms of ( W ) using the volume equation: [ H = \frac{10}{2W^2} = \frac{5}{W^2} ]

Substituting this into the cost function, we get: [ C = 20W^2 + 36W\left(\frac{5}{W^2}\right) ] [ C = 20W^2 + \frac{180}{W} ]

To minimize ( C ), we take its derivative with respect to ( W ) and set it equal to zero: [ \frac{dC}{dW} = 40W - \frac{180}{W^2} = 0 ]

Solving this equation gives us the value of ( W ), which we can then use to find ( L ) and ( H ), and ultimately calculate the minimum cost.

Once we find the values of ( W ), ( L ), and ( H ), we substitute them into the cost function ( C ) to find the minimum cost of materials for the container.

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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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