It's not that easy!

Geometry Level 4

Two identical right circular cones each of height 2 cm are placed vertical apex downward.At the start, the upper cone is full of water and lower cone is empty.Then water drips down through a hole in the apex of upper cone into the lower cone. The cube of height of water in the lower cone at the moment when height of water in upper cone is 1 cm is:

1/7 1/2 -1/2 7 1/4 1/8 -1/4

2 solutions

Chew-Seong Cheong
Mar 2, 2015

The volume of water in an inverted cone is given by $V = \frac{1}{3}\pi r^2 y$ , where $r$ is the base radius and $y$ is the height from the vertex. It is noted that $r \propto y \Rightarrow V(y) \propto y^3$ .

Let the full cone of water be $V_f$ , therefore, $\dfrac {V(y)}{V_f} = \dfrac {y^3}{2^3} \Rightarrow V(y) = \dfrac {y^3}{8} V_f$ .

The water volume of top cone, $V(1) = \frac {1}{8} V_f$ . And the water in the bottom cone $V(y_b) = V_f - \frac {1}{8} V_f = \frac {7}{8} V_f = \dfrac {y_b^3}{8} V_f\Rightarrow y_b^3 = \boxed{7}$

Why is it level 5 sir? I think people got scared because of the title :P !

Venkata Karthik Bandaru - 6 years, 2 months ago

you remind me of hyunh phu so! (check in internet)

avinesh krishnan - 6 years, 3 months ago

Niranjan Khanderia
May 23, 2015

Chew-Seong Cheong ,nice short solution. I did with the same idea but rather in a long way. By using proportionality you made it short.

Thanks Mr. Khanderia, sir (following the Indian tradition calling every one older as sir). You have been very active in Brilliant.org. You are as old as my mom, but she sure can't do math.

Chew-Seong Cheong - 6 years ago

Thanks for your sentiments. I can do a little math. But a mom does many more things useful in life. No comparison. And I feel so happy when I see teens knowing much more then me. With best wishes.

Niranjan Khanderia - 6 years ago

It's not that easy!

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