an open box with a square base have a volume of 8m^3. the material of the base costs $8 per square meter and the material for the sides costs $6 per square meter. express the cost of the materials as a function of the width (w) of the base

(Refer to picture for diagram.)

Area of base $=w^{2}$
Area of one side $=\frac{8}{w^{2}}*w=\frac{8}{w}$

Cost of base material $=8w^{2}$
Cost of material for a side $=6*\frac{8}{w}=\frac{48}{w}$
There are 4 sides so $4*\frac{48}{w}=\frac{192}{w}$

Total cost $C(w)=8w^{2}+\frac{192}{w}$

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MrRoyal MrRoyal · Answer 2 · 2021-10-22T08:31:17-04:00

The volume of a box is the amount of space in it.

The cost function is: $\mathbf{C(w)=8w^2 + \frac{192}{w}}$

The volume of the box is:

$\mathbf{V = 8}$

Assume the base length is w.

So, the area of the base is:

$\mathbf{A_1 = w \times w}$

$\mathbf{A_1 = w^2}$

The base material costs $8 per square meter.

So, the cost of the base material would be:

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Recall that:

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The volume of a box with a square base is:

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Make h the subject

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Substitute 8 for V

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The side area is:

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So, we have:

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The side materials cost $6 per square meter.

So, the cost of the side materials would be:

$\mathbf{C_2 = 6 \times \frac{32}{w}}$

$\mathbf{C_2 = \frac{192}{w}}$

The cost function of the box is:

$\mathbf{C =C_1 + C_2}$

So, we have:

$\mathbf{C=8w^2 + \frac{192}{w}}$

Hence, the cost function is:

$\mathbf{C(w)=8w^2 + \frac{192}{w}}$