Tilted Drum

Geometry Level 4

A cylinder drum with an open top and height of 90 cm \text{90 cm} is filled to the top with 210 L \text{210 L} of water. What is the remaining volume of water (in L \text L ) when the drum is tilted 1 0 10^\circ with the vertical? (Give your answer to the nearest integer.)


The answer is 199.

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3 solutions

Chew-Seong Cheong
Apr 11, 2020

We note that the volume of water spilled out Δ V \Delta V is a half of the volume of a cylinder same radius r r as the drum but with a height of 2 r tan 1 0 2r \tan 10^\circ . In equation,

Δ V = 2 r tan 1 0 2 h V where h and V are the height and volume of the drum. = tan 1 0 1000 V 3 π h 3 Note that r = 1000 V π h = tan 1 0 1000 21 0 3 π 9 0 3 and V = 210 and h = 90 11.2 \begin{aligned} \Delta V & = \frac {2r\tan 10^\circ}{2h} V & \small \blue{\text{where }h \text{ and }V \text{ are the height and volume of the drum.}} \\ & = \tan 10^\circ \sqrt{\frac {1000V^3}{\pi h^3}} & \small \blue{\text{Note that }r=\sqrt{\frac {1000V}{\pi h}}} \\ & = \tan 10^\circ \sqrt{\frac {1000\cdot 210^3}{\pi \cdot 90^3}} & \small \blue{\text{and }V = 210 \text{ and }h=90} \\ & \approx 11.2 \end{aligned}

Therefore the remaining volume of water V Δ V = 210 11.2 199 L V - \Delta V = 210 - 11.2 \approx \boxed{199} \text{ L} .

Typo in the figure. 2 r t a n ( 10 ) \dfrac{2r}{tan(10)}

Niranjan Khanderia - 1 year, 1 month ago

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No, it is correct. h 2 r = tan 1 0 \dfrac h{2r} = \tan 10^\circ h = 2 r tan 1 0 \implies h = 2r \tan 10^\circ . I used this fact in the calculations.

Chew-Seong Cheong - 1 year, 1 month ago

Sorry. I made the mistake because your h and my h are different.

Niranjan Khanderia - 1 year, 1 month ago

H e i g h t o f w a t e r a t t h e o t h e r e n d h 1 = 90 2 r T a n ( 10 ) c m . Height~of~water~at~the~other~end~h_1=90-2r*Tan(10)~cm.\\

r = 210000 90 π = 7000 3 π c m . r=\sqrt{\dfrac{210000}{90*\pi}}=\sqrt{\dfrac{7000}{3*\pi}} ~cm.
S o w a t e r l e f t = π r 2 90 + h 1 2 = 1 / 2 π 7000 / ( 3 π ) ( 180 2 7000 / ( 3 π ) T a n ( 10 ) ) c m 3 199 L So~water ~left=\pi*r^2*\dfrac{90+h_1} 2=1/2*\pi*7000/(3*\pi)*(180-2*\sqrt{7000/(3*\pi)}*Tan(10))~ cm^3\approx 199~ L\\

Vinod Kumar
Apr 11, 2020

V=pi (R^2)[H-2R Tan(x)+R Tan(x)], H=90 R=sqrt[0.21 10^6/(H*pi)]=27.253, x=10 degree V=198.78, Answer=199

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