Question

A cone height 8 m has a curved surface area 188.4 square meters find its volume.(take π=3.14).

Answer 1

Given, height of the cone h = 8

Let r is the radius and l is the slant height of the cone.

Given curved surface area of the cone = 188.4

=> πrl = 188.4

=> πr√(r2 + h2 ) = 188.4             {since slant height l = √(r2 + h2 )}

=> 3.14*r√(r2 + 82 ) = 188.4

=> r√(r2 + 64 ) = 188.4/3.14

=> r√(r2 + 64 ) = 60

=> r2 *(r2 + 64 ) = (60)2

=> r4 + 64r2 = 3600

=> r4 + 64r2 - 3600 = 0

=> (r2 - 36)*(r2 + 100) = 0

=> r2 = 36, -100

Since square of a number can not be negative.

Hense, r2 = -100 is not possible

So, r2 = 36

=> r = ±6

Again since radius can not be negative.

So, r = 6

Now volume of the cone = (1/3)*πr2 h

                                  = (1/3)*π*62h

                                  = (3.41*36*8)/3

                                  = 3.41*12*8

                                  = 301.44 cm3

Answer 2

Given :-

• If a cone of height 8 M has a curved surface area of 188.4 m²

To Find :-

• Radius

• Volume

Solution :-

We know that

$\longrightarrow$ l² = r² + h²

$\longrightarrow$ l² = r² + (8)²

$\longrightarrow$ l² = r² + 64

$\longrightarrow$ l = √(r² + 64)

Now,

$\implies$188.4 = 3.14 × r × √(r² + 64)

$\implies$188.4/3.14 = r × √(r² + 64)

$\implies$60 = r × √(r² + 64)

Squaring both side

(60)² = [r × √(r² + 64)]²

3600 = r²(r² + 64)

3600 = r⁴ + 64r²

$\longrightarrow$ 0 = r⁴ + 64r² - 3600

$\longrightarrow$ 0 = r⁴ + (100r² - 36r²) - 3600

$\longrightarrow$ 0 = r⁴ + 100r - 36r² - 3600

$\longrightarrow$ 0 = (r² + 100)(r² - 36)

Either

$\implies$ r² = -100

or

$\implies$ r² = 36

As length can't be negative

$:\implies$ r² = 36

$:\implies$ r = √36

$:\implies$ r = 6

Now

$\longrightarrow$Volume of cone = 1/3 × π × r² × h

$\longrightarrow$Volume = 1/3 × 22/7 × (6)² × 8

$\longrightarrow$Volume = 22/21 × 36 × 8

$\longrightarrow$Volume = 22/7  × 12 × 8

$\longrightarrow$Volume = 2112/7

$\longrightarrow$Volume = 301.7 m³