Abstract Ice Cream Cone

Geometry Level 3

Three balls are placed inside a cone such that each ball is in contact with the edge of the cone and the next ball. If the radii of the balls are 20 cm, 12 cm, and r r cm from top to bottom, what is the value of r r ?

6.28 8 5 7.2

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Abdulrahman El Shafei by Abdulrahman El Shafei , 23, Egypt

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4 solutions

Discussions for this problem are now closed

Kenny Lau
Dec 6, 2014

image image

There are three similar right-angled triangles in the picture, and from the diagram, x r = x + r + 12 12 = x + r + 44 20 \displaystyle\frac xr=\frac{x+r+12}{12}=\frac{x+r+44}{20}

Solve the equations and we have r = 7.2 r=7.2 .

87 Helpful 4 Interesting 7 Brilliant 3 Confused

Nice answer :-)

Abdulrahman El Shafei - 6 years, 5 months ago
Nada Zohiery
Dec 7, 2014

20/12=12/r then r=(12*12)/20=7.2

58 Helpful 3 Interesting 4 Brilliant 4 Confused

please, explain your point of view. (how you get this relation without using similarity)

Abdulrahman El Shafei - 6 years, 5 months ago

by the AAA similarity the three triangle formed by the 3 different radii.

Afreen Sheikh - 6 years, 4 months ago

I know she should use similarity to get r r , but she didn't mention that at all.

Abdulrahman El Shafei - 6 years, 4 months ago

Very simply, What a clever.!

RungAroon Kaewmanee - 6 years, 6 months ago

best answer.u r clever

Mrunalini Garud - 6 years, 6 months ago
Nihar Mahajan
Dec 21, 2014

12 is geometric mean of 20 and r thus 12 = (20 * r)^(1/2)

144 = 20 * r

r = 144/20

r=7.2

24 Helpful 1 Interesting 1 Brilliant 2 Confused

Create a triangle with legs 12 + r 12+r and 12 − r 12-r from the bottom to the middle circle. Do the same from the middle to the top circle. Now notice that they are similar so 12 + r 12 − r = 20 + 12 20 − 12 \dfrac{12+r}{12-r}=\dfrac{20+12}{20-12} 5 r = 36 = 7.2 5r=36=\boxed{7.2}

15 Helpful 1 Interesting 1 Brilliant 0 Confused

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