1. ## surface area

A quick question:

Cans of soup are often packed in boxes with spaces in between. Calculate the area that is wasted in between all the cans.

The length is 24 cm and the width is 18 cm.

2. You didn't mention what the radius of the circle of each can was so let's just call it r. I'm assuming we're looking from an overhead point of view and we're just focusing on the area of the 2-D plane created by the circles of the cans and the rectangle of the box.

If the cans are positioned and aligned perfectly, the length of the rectangle should equal to the sums of the diameters of each can running along the length. So, if the radius of the circles is equal to r, then the diamater of each circle is 2r.

The number of cans that can fit along the length is $\displaystyle \frac{24 cm}{2r cm} = \frac{12}{r}$ (for example, if the diameter of each circle was 12 cm, then you can only fit 24/12 = 2 cans across the length of the box).

Similar reasoning shows that there'll be $\displaystyle \frac{18}{2r} = \frac{9}{r}$ cans along the width of the box.

So, looking at the entire box, you'll see that there'll be $\displaystyle \frac{9}{r} \times \frac{12}{r}$ circles, each with an area of $\displaystyle \pi r^{2}$. So you can find the summed area made by the cans and subtract it from the area of the rectangle created by the box.