The terms of a sequence, defined by the iterative formula $x_{n+1}=\ln \left(x_n^2+4\right),$ converge to the value $\alpha .$ The first term of the sequence is $2.$

The value $\alpha$ is a root of an equation of the form $x^2=f\left(x\right).$ Find this equation.

The equation $\frac{\sin (x-1)}{2x-3}+1=0$ has a root, $\alpha ,$ between $x=1$ and $x=1.4.$

Show that $\alpha$ also satisfies the equation $x=\frac{3-\sin (x-1)}{2}.$

The equation $\frac{\sin (x-1)}{2x-3}+1=0$ has a root, $\alpha ,$ between $x=1$ and $x=1.4.$

Root $\alpha$ also satisfies the equation $x=\frac{3-\sin (x-1)}{2}$.

Using an iterative formula based on the equation $x=\frac{3-\sin (x-1)}{2}$ with a suitable starting value, find the value of $\alpha$ correct to $3$ significant figures. Give the result of each iteration to $5$ significant figures.

The graphs of $y={\left(\frac{3}{2}\right)}^x-1$ and $y=x$ intersect at the points $O(0,0)$ and $A.$

Sketch these graphs on the same diagram.

The graphs of $y={\left(\frac{3}{2}\right)}^x-1$ and $y=x$ intersect at the points $O(0,0)$ and $A.$

Using logarithms, find a suitable iterative formula that can be used to find the coordinates of the point $A.$

The graphs of $y={\left(\frac{3}{2}\right)}^x-1$ and $y=x$ intersect at the points $O(0,0)$ and $A.$

Calculate the length of the line $OA,$ giving your answer correct to $2$ significant figures . Give the value of any iterations you calculate to a suitable number of significant figures.

The diagram shows a container in the shape of a cone with a cylinder on top.

The height of the cylinder is $3$ times its base radius, $r.$

The volume of the container must be $5500{ \mathrm{cm}}^3.$ The base of the cone has a radius of $r \mathrm{cm}.$

Write down an expression for the height of the cone in terms of $r.$

The diagram shows a container in the shape of a cone with a cylinder on top.

The height of the cylinder is $3$ times its base radius, $r.$

The volume of the container must be $5500{ \mathrm{cm}}^3.$ The base of the cone has a radius of $r \mathrm{cm}.$

Show that $2\mathrm{\pi }{r}^3+11\mathrm{\pi }{r}^2-5500=0$.

The diagram shows a container in the shape of a cone with a cylinder on top.

The height of the cylinder is $3$ times its base radius, $r.$

The volume of the container must be $5500{ \mathrm{cm}}^3.$ The base of the cone has a radius of $r \mathrm{cm}.$

Show that $2\mathrm{\pi }{r}^3+11\mathrm{\pi }{r}^2-5500=0$.

The equation $2\mathrm{\pi }{r}^3+11\mathrm{\pi }{r}^2-5500=0$ has a root $\alpha$.

Show that $\alpha$ is also a root of the equation $r=\sqrt[3]{\frac{5500-11\pi r^2}{2\pi }}$