The volume of the frustum of a cone is πh/3[r₁² + r₂² - r₁r₂], where h is vertical height of the frustum and r₁, r₂ are the radii of the ends. Is the following statement true or false and justify your answer

Solution:

Given, the radii of frustum of a cone are r₁ cm and r₂ cm.

Vertical height of the frustum cone is h cm.

We have to determine if the volume of the frustum cone is πh/3[r₁² + r₂² - r₁r₂]

Consider a frustum of a cone with slant height ‘l’ and height ‘h’

We know that cross sectional area of the cone = (1/3)πh[r₁² + r₂² + r₁r₂]

Extend sides AD and BC to meet at O.

Frustum of a cone is the difference of two right circular cones OAB and OCD.

Let h₁ and l₁ be the eight and slant height of cone OAB

Let h₂ and l₂ be the height and slant height of cone OCD

Considering triangles OPA and OQD,

As the cones are right circular, ∠OPA = ∠OQD = 90°

∠POA = ∠QOD = common

By AAA criteria, the triangles OPA and OQD are similar.

By the property of similar triangles,

The corresponding sides are proportional

PA/QD = OA/DO = PO/OQ

From the figure,

r₁/r₂ = h₁/h₂ = l₁/l₂

So, r₁/r₂ = h₁/h₂ (or) r₂/r₁ = h₂/h₁

Subtracting 1 from both sides,

r₁/r₂ - 1= h₁/h₂ - 1

(r₁ - r₂)/r₂ = (h₁ - h₂)/h₂

From the figure,

h₁ - h₂ = h

(r₁ - r₂)/r₂ = h/h₂

h₂ = r₂h/(r₁ - r₂) ------------------------- (1)

Considering r₂/r₁ = h₂/h₁

Subtracting 1 from both sides,

r₂/r₁ - 1= h₂/h₁ - 1

(r₂ - r₁)/r₁ = (h₂ - h₁)/h₁

(r₁ - r₂)/r₁ = (h₁ - h₂)/h₁

(r₁ - r₂)/r₁ = h/h₁

h₁ = r₁h/(r₁ - r₂) --------------------------- (2)

Volume of frustum cone = volume of cone OAB - volume of cone OCD

Volume of cone OAB = (1/3)r₁²h₁

Volume of cone OCD = (1/3)r₂²h₂

Volume of frustum cone = (1/3)r₁²h₁ - (1/3)r₂²h₂

Substituting (1) and (2),

= π/3[r₁²(r₁h/(r₁ - r₂)) - r₂²(r₂h/(r₁ - r₂))]

= πh/3 [(r₁³ - r₂³)/(r₁ - r₂)]

By using algebraic identity,

(a³ - b³) = (a - b)(a² + ab + b²)

= πh/3[(r₁ - r₂)(r₁² + r₁r₂ + r₂²)/(r₁ - r₂)]

= πh/3[r₁² + r₁r₂ + r₂²]

Therefore, the volume of frustum cone is πh/3[r₁² + r₁r₂ + r₂²]

✦ Try This: The radii of the ends of a frustum of a cone 20 cm high are 7 cm and 9 cm. Find the slant height and volume of the frustum of a cone.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13

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NCERT Exemplar Class 10 Maths Exercise 12.2 Problem 5

The volume of the frustum of a cone is πh/3[r₁² + r₂² - r₁r₂], where h is vertical height of the frustum and r₁, r₂ are the radii of the ends. Is the following statement true or false and justify your answer

Summary:

The statement “The volume of the frustum of a cone is πh/3[r₁² + r₂² - r₁r₂], where h is vertical height of the frustum and r₁, r₂ are the radii of the ends.” is false

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