Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
In the diagram, $A$ lies on the circumference of a circle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts the circumference at $B$ and $C$. The angle $OAB$ measures $x$ radians. The shaded region is enclosed by $AB$, $AC$ and the circular arc with centre $A$ joining $B$ to $C$. The perimeter of the shaded region is half the circumference of the circle.
(i)[3]
Show that $x = \cos^{-1}\left(\frac{\pi}{4 + 4x}\right)$.
(ii)[2]
Verify by calculation that $x$ lies between $1$ and $1.5$.
(iii)[3]
Use the iterative formula $x_{n+1} = \cos^{-1}\left(\frac{\pi}{4 + 4x_n}\right)$ to find $x$ correct to $2$ decimal places. Record each iterate to $4$ decimal places.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use correct arc formula and form an equation in $r$ and $x$” …