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1 day ago

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A solid cylinder is radiating power. It has a length that is ten times its radius. It is cut into a number of smaller cylinders,

each of which has the same radius. Each small cylinder has the same temperature as the original cylinder. The total radiant power emitted by the pieces is twice that emitted by the original cylinder. How many smaller cylinders are there? Give your answer as a number with no units.

1 answer:

Ostrovityanka [2.2K]1 day ago

3 0

Answer:

The total count of small cylinders equals 7.

Explanation:

Let

the radius of the large cylinder equal R

and the length as L

L = 10 R

The overall area A = 2 π R² + π R L

The length of individual small cylinders is denoted as l

The number of small cylinders is n

L = n l

The total surface area of the smaller cylinders

A'=n (2 π R² + π R l)

Given that emissions power can be expressed as

P = A ε σ T⁴

For the large cylinder

P = A ε σ T⁴ -----------1

For the smaller cylinders

P'=A' ε σ T⁴ ------2

From 1 and 2

Given that

P'= 2 P

A' ε σ T⁴ =2 A ε σ T⁴

A'=2 A (Other factors remain constant)

n (2 π R² + π R l) =(2 2 π R² + π R L)

n (2 R² + R l) = (2 R² + R L)

$n(2R^2+R\times \dfrac{L}{n}) = 2(2R^2+RL)$

L = 10 R

$n(2R^2+R\times \dfrac{10R}{n}) =2 (2R^2+R\times 10R)$

$n(2+\dfrac{10}{n}) =2( 2+ 10)$

2 n +10 = 2 x 12

2 n +10 = 24

2 n = 24 -10

2 n = 14

n = 7

The total number of small cylinders equals 7.

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