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The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder, if its total surface area is $616 cm 2$ .

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2565

c m 3

2987

c m 3

2056

c m 3

1078

c m 3

answer is D.

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Detailed Solution

The ratio between the curved surface area and the total surface area of a right circular cylinder is

1 : 2

.
Let r be the radius of the base of the cylinder and h be the height of the cylinder.
Total surface area =

2 π r h + r

Curved surface area =

2 π r h

Volume =

π r 2 h

So,

2 π r h : 2 π r h + r = 1 : 2

⇒ 2 π r h 2 π r h + r = 12

⇒ h h + r = 12

⇒

2 h = h + r

⇒

h = r

… … (1)

Since, the total surface area of the cylinder is

616 c m 2

.
So,

2 π r h + r = 616.... (2)

Put (1) in (2),

2 π r r + r = 616 ⇒ 2 π r × 2 r = 616 ⇒ 2 × 227 × 2 × r 2 = 616 ⇒ 887 r 2 = 616

⇒ 788 × 887 × r 2 = 788 × 616 ⇒ r 2 = 49 ⇒ r 2 = 72 ⇒ r = 7 c m

Volume of the cylinder =

π r 2 h

⇒ V o l u m e o f t h e c y l i n d e r = 227 × 72 × 7 ⇒ V o l u m e o f t h e c y l i n d e r = 22 × 7 × 7 ⇒ V o l u m e o f t h e c y l i n d e r = 1078 c m 3

Therefore, the volume of the cylinder is

1078 c m 3

.
Hence option (4) is correct.

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