Question

The curved surface area of a cone is 20 cm and its
slant height and radius are consecutive natural
numbers. Then, its volume equals
(1) 161 cm3
(2) 321 cm3
(3) 201 cm3
(4) 81 cm3​

Answer 1

{ Curved surface area be 20π cm² }

Answer:

  Volume of the cone is 16π cm³

Step-by-step explanation:

Let the radius of the base and slant height of the cone are x cm and (x + 1) cm respectively.

By the given condition,

    curved surface area = 20π cm²

or, π × radius of the base × slant height = 20π

or, π x (x + 1) = 20π

or, x² + x = 20

or, x² + x - 20 = 0

or, (x + 5) (x - 4) = 0

Either x + 5 = 0 or, x - 4 = 0

Since x is a natural number, x = 4

So radius of the base = 4 cm

    and slant height = 4 + 1 cm = 5 cm

Then the height of the base (h)

= √(5² - 4²) cm = 3 cm

Hence the volume of the cone is

  = 1/3 × π × (radius of the base of the cone)² × h

  = 1/3 × π × 4² × 3

  = 16π cm³

Answer 2

Given :

( Curved surface area be 20π cm² )  

Step-by-step explanation:

Let the radius of the base and slant height of the cone are x cm and (x + 1) cm respectively.

By the given condition, curved surface area = 20π cm²

or, π × radius of the base × slant height = 20π

or, π x (x + 1) = 20π

or, x² + x = 20

or, x² + x - 20 = 0

or, (x + 5) (x - 4) = 0

Either x + 5 = 0 or, x - 4 = 0

Since x is a natural number, x = 4

So radius of the base = 4 cm  and slant height = 4 + 1 cm = 5 cm

Then the height of the base (h)  = √(5² - 4²) cm = 3 cm

Hence the volume of the cone is,

= 1/3 × π × (radius of the base of the cone)² × h

= 1/3 × π × 4² × 3

= 16π cm³

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