# Similar shapes (Volume and surface area)

• Mar 26th 2011, 05:20 AM
Natasha1
Similar shapes (Volume and surface area)
Oil is stored in either small drums or large drums. The shapes of the drums are mathematically similar.

A small drum has a volume of 0.006m3 and a surface area of 0.2m2

The height of a large drum is 3 times the height of a small drum.

1) Calculate the volume of a large drum?

How do I go about doing this as I know it's not 3 x 0.006

2) The cost of making a drum is £1.20 for each m2 of surface area. A company wants to store 3240m3 of oil in large drums. Calculate the cost of making enough large drums to store this oil?

Any help would be much appreciated.
• Mar 26th 2011, 05:30 AM
emakarov
Let h, s, v be the height, surface area and volume, respectively, of the small drum, and let H, S, V be the height, surface area and volume of the large drum. Since the drums are similar, $\displaystyle S/s = (H/h)^2$ and $\displaystyle V/v=(H/h)^3$. Since you know H/h, s and v, you can find S and V.
• Mar 26th 2011, 05:37 AM
Natasha1
Why do we have to square H/h?
• Mar 26th 2011, 05:42 AM
Natasha1
1) V = 0.006 x 27 = 0.162m3

2) 3240 / 0.162 = 20,000 large drums

so 20,000 x 1.2 = £24,000
• Mar 26th 2011, 05:53 AM
emakarov
That's a property of similar figures. You can prove it in some specific cases. For example, suppose drum are cylinders, the diameter of the small drum is r and that of the large drum is R. By the definition of similarity, R / r = H / h. Let's call this ratio k. Then $\displaystyle S = 2\pi R^2 + 2\pi R H$ (top and bottom circles plus the side). So, $\displaystyle S = 2\pi(kr)^2+2\pi(kr)(kh)=k^2(2\pi r^2+2\pi rh)=k^2s$. The principle is true for arbitrary similar figures, though. One probably needs calculus to prove it in general.
• Mar 26th 2011, 05:56 AM
emakarov
Quote:

so 20,000 x 1.2 = £24,000
You need to multiply £1.2 by the total number of square meters, not by the number of drums. For this, you need to find the surface area S of the large drum and multiply it by 20,000.