The constant \(a\) is such that \(\int_{1}^{a} x^2 \ln x \, dx = 4\).

(a) Show that \(a = \left( \frac{35}{3 \ln a - 1} \right)^{\frac{1}{3}}\).

(b) Verify by calculation that \(a\) lies between 2.4 and 2.8.

(c) Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.