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Math Help - Similar shapes (Volume and surface area)

  1. #1
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    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.
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  2. #2
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    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, S/s = (H/h)^2 and V/v=(H/h)^3. Since you know H/h, s and v, you can find S and V.
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  3. #3
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    Why do we have to square H/h?
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  4. #4
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    1) V = 0.006 x 27 = 0.162m3

    2) 3240 / 0.162 = 20,000 large drums

    so 20,000 x 1.2 = 24,000
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  5. #5
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    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 S = 2\pi R^2 + 2\pi R H (top and bottom circles plus the side). So, 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.
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  6. #6
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    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.
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