Author Topic: Ratio (Read 578 times) Tweet Share

kenhung123 · « **on:** June 16, 2010, 02:57:16 pm »

If 2 similar cones of a vertical height of 5 cm and 10cm respectively
What would be their ratio of areas?

I did this:
(1/2)^3=0.125

125:1000
1:8

brightsky · « **Reply #1 on:** June 16, 2010, 06:41:35 pm »

Let $x$ cm be the radius of the base of the smaller cone. Then the "slant" of this cone would be $\sqrt{5^2 + x^2} = \sqrt{25 + x^2}$

Because the cones are similar, then the other cone would have radius $2x$ . Then the "slant" of this cone would be $\sqrt{10^2 + 4x^2} = \sqrt{100 + 4x^2} = 2\sqrt{25 + x^2}$

The total surface area of the first cone would be $\pi \cdot x^2 + \pi \cdot x \cdot \sqrt{25 + x^2}$ .

The total surface area of the second cone would be $\pi \cdot 4x^2 + \pi \cdot 2x \cdot 2 \sqrt{25 + x^2}$ .

Dividing the first over the second, you get 1/4.

Yitzi_K · « **Reply #2 on:** June 16, 2010, 06:55:11 pm »

There's a far easier way...

The enlargement factor, $k$ , between the two cones is 2, as one has twice the dimensions of the other.

The ratio of the areas of 2 similar shapes is k squared, so the answer is $2^2=4$ , so the ratio is 1:4

TyErd · « **Reply #3 on:** June 16, 2010, 06:59:24 pm »

Quote from: Yitzi_K on June 16, 2010, 06:55:11 pm

There's a far easier way...

The enlargement factor, $k$ , between the two cones is 2, as one has twice the dimensions of the other.

The ratio of the areas of 2 similar shapes is k squared, so the answer is $2^2=4$ , so the ratio is 1:4

Yes I like this way, did it in further. kenhung123 you would be correct if it asked for the volume.

Just to add, my working out would be:

$k = 5:10 = 1:2$

$k^2 = 25:100 = 1:4$

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