Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The figure shows $A$ at the midpoint of the semicircle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts the semicircle at $B$ and $C$. The angle $OAB$ is $x$ radians. The shaded region enclosed by $AB$, $AC$ and the arc centred at $A$ has area equal to one half of the semicircle.
(i)[1]
From triangle $OAB$, establish that $AB = 2r\cos x$.
(ii)[2]
Therefore show that $x = \cos^{-1}\!\left(\sqrt{\frac{\pi}{16x}}\right)$.
(iii)[2]
Confirm by calculation that $x$ lies between $1$ and $1.5$.
(iv)[3]
Use an iteration based on the equation in part (ii) to find $x$ correct to $3$ decimal places. Record each iterated value to $5$ decimal places.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Trigonometry applied correctly to deduce $AB=2r\cos x$” …