Question

A rectangular storage container with an open top has a volume of 14 m3. The length of its base is twice its width. Material for the base costs $5 per square meter; material for the sides costs $4 per square meter. Express the cost of materials as a function of the width of the base.

Answer 1

Answer:

$Total cost=10x^{2} dollars+\frac{168dollars}{x}$

Explanation:

First at all, let's see the figure in the attachment where we define:

Width is "x", Lenght must be "2x" and Height is "h".

The volume of a rectangular storage container is defined as:  

V=lengh*width*height

So, replacing values, we have:

V=$2x^{2} h$

Considering that V=14m3 we can clear "h" in function of "x" (the width):

$V=2x^{2} h=14; h=\frac{7}{x^{2}}$

Now, we calculate the areas of the container:

$Ax_{1} =x*h\\Ax_{2}=2x*h\\Ax_{3}=2x*x$

Where: Ax1 is the side 1 area; Ax2 is the side 2 area and Ax3 is the base area

Replacing "h" on the previous equations, we have:

$Ax_{1}=x*h=x*\frac{7}{x^{2} }=\frac{7}{x}\\Ax_{2}=2x*h=2x*\frac{7}{x^{2} }=\frac{14}{x}\\Ax_{3}=x*2x=2x^{2}$

Remember that the container is open at the top, so we have to calculate just one area in the base. The sides 1 and 2 are 2 of each one.

So, we have: Total area = 2*Ax1 + 2*Ax2 + Ax3

Now for the total cost of materials, we have: Total cost=Cost (2*Ax1) + Cost (2*Ax2) + Cost (Ax3)

For the sides 1 and 2, we have a cost of:

$Cost Ax_{1}=\frac{7}{x}m^{2} *\frac{4 dollars}{m^{2} }=\frac{28dollars}{x}\\Cost Ax_{2}=\frac{14}{x}m^{2} *\frac{4 dollars}{m^{2} }=\frac{56dollars}{x}\\Cost Ax_{3}=(2x^{2})m^{2}*\frac{5 dollars}{m^{2}}=10x^{2}dollars$

Finally, total cost is:

$Total cost=2*\frac{28 dollars}{x} + 2*\frac{56 dollars}{x} + 10x^{2} dollars\\Total cost=10x^{2} dollars+\frac{168dollars}{x}$