By sketching a suitable pair of graphs, show that the equation
$$3 \ln x = 15 - x ^ { 3 }$$
has exactly one real root.
Show by calculation that the root lies between 2.0 and 2.5.
Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.