Here is my step-by-step-work. Let me know if you have any questions! :)
Answer:
The lifetime of the particle is 
Explanation:
From the question we are told that
The average rest energy is 
The intrinsic width is 
The lifetime is mathematically represented as

Where h is the Planck's constant with a value of
substituting values


Refer to the diagram shown below
Given:
m = 2.8 kg, the mass of the bucket
a =3.1 m/s², the acceleration of the bucket
Therefore
W = 2.8*9.8 = 27.44 N, the weight of the bucket.
Let T = the tension in the rope.
From the free body diagram, the net force accelerating the bucket is
T - W = m*a
That is,
T = W + m*a
= 27.44 + 2.8*3.1 N
= 36.12 N
Answer: d. 36.1 N