Question

The curved surface area of a cone is 20cm^2 and its slant height and radius are consecutive natural no.s then find its volume

Answer 1

Answer:

Volume of the cone is 10.23 cm³

Step-by-step Solution:

Let the 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 = 20 cm²

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

or, π x (x + 1) = 20

or, 22/7 × (x² + x) = 20

or, 11 (x² + x) = 7 × 10

or, 11x² + 11x = 70

or, 11x² + 11x - 70 = 0

∴ x = {- 11 ± √(11² + 3080)} / 22

≈ (- 11 ± 56.58) / 22

= 2.07 , since x > 0

So radius of the base = 2.07 cm

and slant height = 2.07 + 1 = 3.07 cm

Then the height of the base (h)

= √(3.07² - 2.07²) cm

= 2.27 cm

Hence the volume of the cone is

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

= 1/3 × 22/7 × 2.07² × 2.28

= 10.23 cm³

Note: Radius of the base of a cone cannot be greater than the slant height, so we change the order of them mentioned in the question. The radius of the base and slant height aren't coming in Natural numbes; there's something wrong with the given data.

Answer 2

Answer:

Volume of cone = 16π cm³

Step-by-step explanation:

Correct Question is The curved surface area of a cone is 20π cm^2

slant height and radius are consecutive natural no.s

as slant height > radius

=> radius = n   & slant height = n+1

Curved surface Area = π * radius * salnt height

= π * n * (n + 1)

π * n * (n + 1)  = 20π

=> n(n + 1) = 20

=> n² + n - 20 = 0

=> n² + 5n - 4n - 20 = 0

=> n(n + 5) - 4(n+5) = 0

=> (n -4)(n+5) = 0

=> n = 4

Radius = 4

Slant height = 5

Height = √5² - 4² = √25 - 16 = √9 = 3

Volume of cone = (1/3)πr²h

= (1/3) π (4)² * 3

= 16π cm³

Volume of cone = 16π cm³