Discussion

Question:

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?

Explanation:

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Applying the concept of similarity,

$\frac{10}{26}=\frac{h}{h+15}$

$\Rightarrow h=9.375 \mathrm{cm}$

Volume of water that overflows can be given as,

$\frac{1}{3}\pi \left[\right({13}^2\times (15+9.375))-(5^2\times 9.375\left)\right]$

$\Rightarrow \frac{1}{3}\pi \left[\right({13}^2\times 24.375)-(5^2\times 9.375\left)\right]$

$\Rightarrow \frac{1}{3}\pi [4119.375-234.375]$

$\Rightarrow 1295\pi {\mathrm{cm}}^3$

Since the radius of the hole is 5 mm i.e. 0.5cm,

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The volume of water flowing in 1 minute,

(0.5² × 1000)π cm³​​​​​​​

Hence, the required time can be given as,

$\frac{1295\pi }{0.5^2\times 1000\pi }=5.18$

Hence, option (d).

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