(i) To find the probability that a fir tree has a height less than 45 metres, we standardize the variable using the formula:

\(Z = \frac{X - \mu}{\sigma}\)

where \(X = 45\), \(\mu = 40\), and \(\sigma = 8\).

\(Z = \frac{45 - 40}{8} = 0.625\)

We find \(P(Z < 0.625)\) using standard normal distribution tables or a calculator, which gives:

\(P(X < 45) = 0.734\)

(ii) To find the probability that a fir tree has a height within 5 metres of the mean, we calculate:

\(P(35 < X < 45) = P(X < 45) - P(X < 35)\)

Using the result from part (i), \(P(X < 45) = 0.734\).

Standardize for \(X = 35\):

\(Z = \frac{35 - 40}{8} = -0.625\)

\(P(Z < -0.625) = 1 - P(Z < 0.625) = 1 - 0.734 = 0.266\)

Thus, \(P(35 < X < 45) = 0.734 - 0.266 = 0.468\)