Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Numerical Solutions of Equations, Exercise 2: EXERCISE 6A

 Verify that one of these roots, $\alpha ,$ lies between $2.1$ and $2.2 .$

Show graphically that the equation $cosecx=\sin x$ has exactly two roots for $0<x<2\pi$.

Using an algebraic method to find the value of the larger root correct to$3$ significant figures.

The equation $f\left(x\right)=20x^3+8x^2-7x-3=0$ has exactly two roots. Without factorising the cubic equation, show, by calculation, that one of these roots is between $0.5$ and $1 .$

If $f\left(x\right)=20x^3+8x^2-7x-3$. Show, by sketching the graph of $y=f\left(x\right)$ for $-1⩽x⩽1,$ that the smaller root is between $-1$ and $0$.

$f\left(x\right)=20x^3+8x^2-7x-3$

Explain why it is not necessary to use a numerical method to find the two solutions of this equation.

A guitar tuning peg is in the shape of a cylinder with a hemisphere at one end. The cylinder is $20 \mathrm{mm}$ long and the whole peg is made from $800{ \mathrm{mm}}^3$ of plastic. The base radius of the cylinder is $r \mathrm{mm}.$

Show that $\pi r^3+30\pi r^2-1200=0$

A guitar tuning peg is in the shape of a cylinder with a hemisphere at one end. The cylinder is $20 \mathrm{mm}$ long and the whole peg is made from $800{ \mathrm{mm}}^3$ of plastic. The base radius of the cylinder is $r \mathrm{mm}.$

Show that the value of $r$ is between $3 \mathrm{mm}$ and $4 \mathrm{mm}.$