Question

A sector of a circle of radius 15 cm has the angle 120º. It is rolled up so that two bounding radii are joined together to form a cone. Find the volume of the cone. `(π = 22/7)`

Answer 1

1. Identify slant height

When a sector of a circle is rolled up to form a cone, the radius of the sector becomes the slant height `(l)` of the cone:

`l` = 15 cm

2. Determine base radius

The arc length of the sector becomes the circumference of the circular base of the cone (2πr).

First, calculate the arc length of the sector:

Arc Length = `θ/(360^circ) xx 2πR`

Arc Length = `(120^circ)/(360^circ) xx 2π xx 15`

= `1/3 xx 30π`

= 10 πcm

Now, equate this to the base circumference of the cone to find its radius (r):

2πr = 10π

r = 5 cm

3. Calculate cone height

Using the Pythagorean theorem in the right-angled triangle formed by the height (h) radius (r) and slant height `(l)`:

`h = sqrt(l^2 - r^2)`

`h = sqrt(15^2 - 5^2)`

= `sqrt(225 - 25)`

= `sqrt(200)`

= `10sqrt(2)` cm

Using `sqrt(2) ≈ 1.4142`:

h ≈ 14.142 cm

4. Compute the volume

The volume (V) of a cone is given by the formula:

`V = 1/3 πr^2h`

Substitute the given value `π = 22/7` along with r = 5 and `h = 10sqrt(2)`:

`V = 1/3 xx 22/7 xx 5^2 xx 10sqrt(2)`

`V = 1/3 xx 22/7 xx 25 xx 10sqrt(2)`

`V = (5500sqrt(2))/21 cm^3`

Approximating the final value:

`V ≈ (5500 xx 1.4142)/21`

V ≈ 370.39 cm³