Question

Two steel sheets each of length a₁ and breadth a₂ are used to prepare the surfaces of two right circular cylinder —one having volume v₁ and height a₂ and other having volume v₂ and height a₁. Then,
A. v₁=v₂B. a₂v₁=a₁v₂C. a₁v₁=a₁v₂D. v₁/a₁ = v₂/a₂

Answer 1

The correct answers among the given alternatives are option B, a2v1 = a1v2 and option D, v1 / a1 = v2 / a2.

• Given,

Length of each steel sheet = a₁

Breadth of each steel sheet = a₂

Volume of the first cylinder = v1

Height of the first cylinder = a2

Volume of the second cylinder = v2

Height of the second cylinder = a1

• Now, we know that the volume of a cylinder is given as : π.(radius)².height

• Let the radius of the first cylinder be r, and that of the second cylinder be r'.

• For the first cylinder, height = a2

=> The sheet is folded along its breadth a2 for the first cylinder.

• Therefore, circumference of the base of first cylinder = length of the sheet = a1

Also, circumference of the base = 2πr

=> a1 = 2πr

Or, r = a1 / 2π -(i)

• For the second cylinder, height = a1

=> The sheet is folded along its length a1 for the second cylinder.

• Therefore, circumference of the base of second cylinder = Breadth of the sheet = a2

Also, circumference of the base = 2πr'

=> a2 = 2πr'

Or, r' = a2 / 2π -(ii)

• Now, volume of the first cylinder (v1) = π.r².a2

Or, v1 = π.(a1 / 2π)².a2 [ Since r = a1 / 2π, from eq. (i) ]

Or, v1 = π.(a1²/4π²).a2

Or, v1 = π.a1².a2 / 4π²

Or, v1 = a1².a2 / 4π -(iii)

• Volume of the second cylinder (v2) = π.r'².a1

Or, v2 = π.(a2 / 2π)².a1 [ Since r' = a2 / 2π, from eq. (ii) ]

Or, v2 = π.(a2²/4π²).a1

Or, v2 = π.a2².a1 / 4π²

Or, v2 = a2².a1 / 4π -(iv)

• Dividing eq. (iii) by eq. (iv), we get,

v1 / v2 = (a1².a2 / 4π) / (a2².a1 / 4π)

Or, v1 / v2 = (a1².a2 / 4π) × (4π / a2².a1)

Or, v1 / v2 = a1 / a2

By cross-multiplying, we get,

v1a2 = a1v2 (Option B)

Or, v1 / a1 = v2 / a2 (Option D)

Answer 2

Answer:

your answer is in the attachment

Step-by-step explanation:

hope it's help you XD