The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.
The winning zone is modelled by the function
![straight f open parentheses x close parentheses equals ln space open parentheses 3 x plus 4 close parentheses minus 0.25 x 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The lower boundary of the winning zone has equation![space y equals straight f left parenthesis x right parenthesis comma space space space x comma space y greater or equal than 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The upper boundary of the winning zone has equation ![y equals straight f open parentheses x close parentheses plus 1 comma space space x comma space y greater or equal than 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A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.
![q5-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots](https://cdn.savemyexams.com/cdn-cgi/image/f=auto,width=1920/uploads/2023/07/q5-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots.png)
Using iterative formulas with initial values
and
as appropriate, find the exact distances between which the ball must stop for this player to win Funcball.
Give your answers to two significant figures.