Question

The radius of a sphere is a random number between 2 and 4. What is the expected value of its volume? What is the probability that its volume is at most 36π?​

Answer 1

Given : The radius of a sphere is a random number between 2 and 4.

To Find :  What is the expected value of its volume

probability that its volume is at most 36π

Solution:

Volume of Sphere = (4/3)πR³

The radius of a sphere is a random number between 2 and 4

f(R)  =  1/(4- 2)     for  2 < R  < 4

          0               R other value

$E(R) = \int\limits^4_2 {\frac{4}{3} \pi R^3 } \, \frac{dR}{2}$

=> E (R)  =  (2π/3)  (R⁴/4)  $\left \right ]_2^4$

=>  E (R)  =  (π/6) [4⁴ - 2⁴]

=> E (R)  =  (π/6) [240]

=> E (R)  =   40π  

expected value of its volume  =   40π   cubic units

36π  = (4/3)πR³

=> R = 3

Hence  probability that its volume is at most 36π =  (3 - 2)/ ( 4- 2)

= 1/2

= 0.5

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