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Inverted Cone Water Flow Time

XAT 2014 · Quantitative Ability & Data Interpretation · Hard · Geometry

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Diameter of the base of a water - filled inverted right circular cone is 26 cm. A cylindrical pipe, 5 mm in radius, is attached to the surface of the cone at a point. The perpendicular distance between the point and the base (the top) is 15 cm. The distance from the edge of the base to the point is 17 cm, along the surface. If water flows at the rate of 10 meters per minute through the pipe, how much time would elapse before water stops coming out of the pipe?

  1. A.

    < 4.5 minutes

  2. B.

    ≥ 4.5 minutes but < 4.8 minutes

  3. C.

    ≥ 4.8 minutes but < 5 minutes

  4. D.

    ≥ 5 minutes but < 5.2 minutes

  5. E.

    ≥ 5.2 minutes

Answer

D

Explanation

Base radius R=13R = 13 cm. In upper right triangle, slant length =17= 17, vertical height =15  ⟹  = 15 \implies radius at pipe height =13−172−152=13−8=5= 13 - \sqrt{17^2 - 15^2} = 13 - 8 = 5 cm. Frustum height h=15h = 15 cm. Volume of water above pipe =13πh(R2+r2+Rr)=13π(15)(132+52+13×5)=5π(169+25+65)=1295π cm3= \frac{1}{3} \pi h (R^2 + r^2 + Rr) = \frac{1}{3} \pi (15) (13^2 + 5^2 + 13 \times 5) = 5 \pi (169 + 25 + 65) = 1295 \pi \text{ cm}^3. Flow rate =πrpipe2×v=π(0.5)2×1000=250π cm3/min= \pi r_{pipe}^2 \times v = \pi (0.5)^2 \times 1000 = 250 \pi \text{ cm}^3/\text{min}. Time =1295π250π=5.18= \frac{1295 \pi}{250 \pi} = 5.18 minutes.

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