(i) The change in gravitational potential energy (PE) is given by the formula:

\(\Delta PE = mgh\)

where \(h = x \sin \alpha\). Given \(\sin \alpha = \frac{5}{13}\), we have:

\(\Delta PE = 8 \times g \times x \times \frac{5}{13}\)

Assuming \(g = 9.8 \text{ m/s}^2\), the change in PE is:

\(\Delta PE = \frac{400}{13} x = 30.8x\)

(ii) Using the energy method, the work done against friction is:

\(WD = 15 \times x\)

The initial kinetic energy (KE) is:

\(KE = \frac{1}{2} \times 8 \times 5^2 = 100\)

Using the energy conservation principle:

\(KE_{\text{initial}} = \Delta PE + WD\)

Substitute the values:

\(100 = \frac{400}{13} x + 15x\)

Simplifying gives:

\(100 = \left( \frac{400}{13} + 15 \right) x\)

\(100 = \frac{260}{13} x\)

Solving for \(x\):

\(x = \frac{260}{119} = 2.18\)