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Two cylinders with equal volume will always have equal surface areas. Is the given statement true or false

Solution:

Let h₁ = height of cylinder 1

Let h₂ = height of cylinder 2

Let r₁ = radius of first cylinder

Let r₂ = radius of second cylinder

We are given that volumes of the cylinder are equal. Therefore

πr₁²h₁ = πr₂²h₂

r₁²h₁ = r₂²h₂

h₁ = r₂²h₂/r₁²

2πr₁h₁ = 2πr₁( r₂²h₂/r₁²) = 2π(r₂²h₂/r₁) = 2πr₂h₂(r₂/r₁)

2πr₁h₁ = Volume of cylinder 1

2πr₂h₂ = Volume of cylinder 2

Since 2πr₁h₁ = 2πr₂h₂(r₂/r₁) it can be stated

Curved Surface area of cylinder 1 = Curved Surface Area of cylinder 2 × (r₂/r₁)

Therefore, if the volume of two cylinders are equal, their curved surface areas are not necessarily equal. The statement is false.

✦ Try This: Two cylinders with equal curved surface areas will always have equal volume. Is the given statement true or false

If curved surface areas of the two cylinders are equal implies:

2πr₁h₁ = 2πr₂h₂

r₁/r₂ = h₂/h₁

The relationship between the volumes of the two cylinders can be established as follows:

Volume of cylinder 1 = πr₁²h₁

Since r₁/r₂ = h₂/h₁

Volume of cylinder 1 = π(h₂r₂/h₁)² h₁

Volume of cylinder 1 = πr₂²h₂ (h₂/h₁)

Volume of cylinder 1 = Volume of cylinder 2 × (h₂/h₁)

Therefore Volumes of cylinder 1 and cylinder 2 are equal only if h₂ = h₁

Hence we can conclude that volumes of cylinder 1 and cylinder 2 are not necessarily always equal.

The statement is false(F).

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 11

NCERT Exemplar Class 8 Maths Chapter 11 Problem 59

Summary:

Two cylinders with equal volume will always have equal surface areas. The statement is False

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