A sphere, a cylinder and a cone have the same radius and same height,

1.	A sphere, a cylinder and a cone have the same radius and same height, then the ratio of their curved surface areas isA) 2 : √3 : 4 B) 4 : 4 : √5 C) 3 : √5 : 4 D) None
Answer» Correct option is: B) 4 : 4 : √5 Given that Radius of cylinder = Radius of cone = Radius of sphere = r \(\because\) Height of the sphere = Diameter of the sphere = 2 r \(\therefore\) Height of cylinder = Height of cone = 2r Now, curved surface area of sphere \(4 \pi r^2\) curved surface are of cylinder = \(2 \pi r h\) = 2\(\pi r (2 r) \) (\(\because\) h = 2r) = \(4 \pi r^2\) curved surface area of cone = \(\pi rl\) = \(\pi r\sqrt{r^2+h^2}\) = \(\pi r\sqrt {r^2 + 4r^2}\) (\(\because\) h = 2r) = \(\sqrt5 \pi r^2\) Now, the ratio of their curved surface areas = \((C.S.A)_S : (C.S.A)_{cylinder} : (C.S.A)_{cone}\) = \(4 \pi r^2\) : \(4 \pi r^2\) : \(\sqrt5 \pi r^2\) = 4 : 4 : \(\sqrt5 \) (On dividing by \(\pi r^2\)) Hence, the ratio for their curved surface areas = 4 : 4 : \(\sqrt5 \). Correct option is: B) 4 : 4 : √5

A sphere, a cylinder and a cone have the same radius and same height, then the ratio of their curved surface areas isA) 2 : √3 : 4 B) 4 : 4 : √5 C) 3 : √5 : 4 D) None

Answer»

Correct option is: B) 4 : 4 : √5

Given that

Radius of cylinder = Radius of cone = Radius of sphere = r

\(\because\) Height of the sphere = Diameter of the sphere = 2 r

\(\therefore\) Height of cylinder = Height of cone = 2r

Now, curved surface area of sphere \(4 \pi r^2\)

curved surface are of cylinder = \(2 \pi r h\)

= 2\(\pi r (2 r) \) (\(\because\) h = 2r)

= \(4 \pi r^2\)

curved surface area of cone = \(\pi rl\)

= \(\pi r\sqrt{r^2+h^2}\)

= \(\pi r\sqrt {r^2 + 4r^2}\) (\(\because\) h = 2r)

= \(\sqrt5 \pi r^2\)

Now, the ratio of their curved surface areas

= \((C.S.A)_S : (C.S.A)_{cylinder} : (C.S.A)_{cone}\)

= \(4 \pi r^2\) : \(4 \pi r^2\) : \(\sqrt5 \pi r^2\)

= 4 : 4 : \(\sqrt5 \) (On dividing by \(\pi r^2\))

Hence, the ratio for their curved surface areas = 4 : 4 : \(\sqrt5 \).

Correct option is: B) 4 : 4 : √5