Ratio of surface areas

Geometry Level 2

The total surface area and the curved surface area of a cylinder that is open at the top are in the ratio 3 : 1 3:1 . What is the ratio of the radius of the cylinder to that of the height?

If your answer is the form of a : b a: b , where a a and b b are coprime positive integers, find a + b a+b .


The answer is 5.

Vignesh Rao by Vignesh Rao , 20, India

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2 solutions

Let r r be the radius of the cylinder and h h be the height. Then, we have

2 π r h + π r 2 2 π r h = 3 1 \dfrac{2\pi rh+\pi r^2}{2\pi rh}=\dfrac{3}{1}

2 π r h + π r 2 = 6 π r h 2\pi rh + \pi r^2 = 6\pi rh

2 r h + r 2 = 6 r h 2rh + r^2 = 6rh

2 h + r = 6 h 2h + r = 6h

r = 4 h r=4h

r h = 4 1 \dfrac{r}{h}=\dfrac{4}{1}

The desired answer is 4 + 1 = 5 4+1=\boxed{5}

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Atanu Ghosh
Feb 28, 2016

T.S.A(except top of the cylinder)/C.S.A=3:1 THEN R:H=4:1 So R+H=5

atanughosh253

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