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An example of a calulus book reads:

A rectangular storage container with an open top has a volume of $10$ m . The length of its base is twice its width. Material for the base costs $10$ per square meter, material for the sides costs $\$6$ per square meter. Express the cost of materials as a function of the width of the base.

We draw the diagram as shown below where $w$ and $2w$ be the width and length of the base $h$ be the height. enter image description here

The area of the base is $(2w)w = 2w^2$, so the cost, in dollars, of the material for the base is $10(2w^2)$. I am able to understand upto this point.

The point which I am unable to understand is where it says two of the sides have area $wh$ and the other two have area $2wh$, so the cost of the material for the sides is $6[2(wh) + 2(2wh)]$. I can take it from here if I get this.

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up vote 1 down vote accepted

$w$ stands for a length in meters. So for example, $w$ could be $3$ meters. You are interested in the area of the base because you want to compute the cost for the base which is the are in square meters times the price per square meter. The price per square meter for the base is $10$ dollars. The area is, as you wrote above, $w \cdot 2w = 2 w^2$. Hence the price for the base is $10 \cdot 2 w^2 = 20 w^2$.

Here you used that the area of a rectangle with sides of lengths $a$ and $b$ respectively is $ a \cdot b$.

Now you try to apply the same for the walls of the box. The $6$ there is again the price for the material per square meter.

Hope this helps. If you have any questions let me know and I'll try to clarify.

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Ok,now i get it.You mean apply area of the rectangle for all 6 sides and add them all up.{sorry for the delayed response.the power went down!} – alok Dec 25 '11 at 14:18
@alok Yes, exactly! : ) But be careful: the box is open and so doesn't have a top and so there are only 5 sides. – Rudy the Reindeer Dec 25 '11 at 14:19

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