(i) We know \(\Sigma (x - k) = 315\) and there are 30 buns, so the mean of \(x - k\) is \(\frac{315}{30} = 10.5\). Since the mean weight of the buns is 50.5 grams, we have:

\(50.5 - k = 10.5\)

Solving for \(k\), we get:

\(k = 50.5 - 10.5 = 40\)

(ii) The variance is given by:

\(\text{var} = \frac{\Sigma (x - k)^2}{30} = \frac{4022}{30} = 134.067\)

We also know the variance formula:

\(\text{var} = \frac{\Sigma x^2}{30} - (\text{mean})^2\)

Substituting the mean \(50.5\), we have:

\(\text{var} = 134.067 - 50.5^2 = 23.817\)

The standard deviation is the square root of the variance:

\(\text{sd} = \sqrt{23.817} = 4.88\)