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Question

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio $$8:5$$, determine the ratio of the radius of the base to the height .

Solution
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$$\cfrac { CSA\quad of\quad cylinder }{ CSA\quad of\quad cone } =\cfrac { 8 }{ 5 } \Rightarrow \cfrac { 2\pi rh }{ \pi rl } =\cfrac { 8 }{ 5 } \Rightarrow 5h=4l=4\sqrt { { r }^{ 2 }+{ h }^{ 2 } } $$
$$\Rightarrow { \left( 5h \right) }^{ 2 }={ \left( 4\sqrt { { r }^{ 2 }+{ h }^{ 2 } } \right) }^{ 2 }\Rightarrow 25{ h }^{ 2 }=16{ r }^{ 2 }+16{ h }^{ 2 }\Rightarrow 9{ h }^{ 2 }=16{ r }^{ 2 }\Rightarrow \cfrac { { r }^{ 2 } }{ { h }^{ 2 } } =\cfrac { 9 }{ 16 } $$
$$\Rightarrow \cfrac { r }{ h } =\sqrt { \cfrac { 9 }{ 16 } } =\cfrac { 3 }{ 4 } =3:4$$

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