(i) To find the gradient of the curve, we need to differentiate \(y = \sqrt{5x + 4}\) with respect to \(x\).

Let \(y = (5x + 4)^{1/2}\).

The derivative \(\frac{dy}{dx} = \frac{1}{2}(5x + 4)^{-1/2} \times 5\).

At \(x = 1\), \(\frac{dy}{dx} = \frac{1}{2}(5 \times 1 + 4)^{-1/2} \times 5 = \frac{5}{6}\).

(ii) We are given \(\frac{dx}{dt} = 0.03\) and need to find \(\frac{dy}{dt}\).

Using the chain rule, \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\).

Substitute \(\frac{dy}{dx} = \frac{5}{6}\) and \(\frac{dx}{dt} = 0.03\):

\(\frac{dy}{dt} = \frac{5}{6} \times 0.03 = 0.025\).