# Lachlan's question via email about the Bisection Method

#### Prove It

##### Well-known member
MHB Math Helper
Perform four iterations of the Bisection Method to find an approximate solution to the equation

$\displaystyle 11\cos{ \left( x \right) } = 1 - 2\,\mathrm{e}^{-x/10}$

when it is known there is a solution in the interval $\displaystyle x \in \left[ 4.65, 4.82 \right]$
The Bisection Method solves equations of the form $\displaystyle f\left( x \right) = 0$ so we must write the equation as $\displaystyle 11\cos{ \left( x \right) } - 1 + 2\,\mathrm{e}^{-x/10} = 0$. We can then see that $\displaystyle f\left( x \right) = 11\cos{ \left( x \right) } - 1 + 2\,\mathrm{e}^{-x/10}$.

I have used my CAS to solve this problem.

After four iterations, we accept $\displaystyle c_4 = 4.68188$ as the root.

I also told the CAS to solve the equation, and this solution is correct to two decimal places, which is very reasonable considering how slowly the Bisection Method converges.