Answer
(a) The mass of sand added to the bucket is 10.6 kg.
(b) $a = 0.88~m/s^2$
Work Step by Step
(a) The system of the block and the bucket will start moving when the weight of the bucket and sand ($m_s ~g$) is greater than the force of static friction acting on the block. Let's find the mass $m_s$ such that $m_s ~g = F_f$. Let $m_b$ be the mass of the block.
$m_s ~g = F_f$
$m_s ~g = m_b ~g ~\mu_s$
$m_s = m_b ~\mu_s = (28.0 ~kg)(0.45) = 12.6 ~kg$
Since the mass of the bucket is 2.00 kg, the mass of sand added to the bucket is 10.6 kg.
(b) We can set up a force equation for the system of the block plus the bucket and the sand. The total mass of the system is $m$ = 28.0 kg + 12.6 kg = 40.6 kg:
$ma = \sum F$
$ma = m_s ~g - m_b ~g ~\mu_k$
$a = \frac{m_s ~g - m_b ~g ~\mu_k}{m}$
$a = \frac{(12.6 ~kg)(9.80 ~m/s^2) - (28.0 ~kg)(9.80 ~m/s^2)(0.32)}{40.6 ~kg}$
$a = 0.88~m/s^2$