Wobbly Table Theorem:

There is a square table with four equal$-$length legs $($see the figure below$),$ where the fourth leg

D$,$ is above the surface, whereas the other three legs A$,$ B$,$ and C are on the uneven surface.

Turn the table around a center so that three legs A$,$ B$,$ and C$,$ stay on the surface.

The distance from leg D to the surface is a function $($f$),$ which depends on the turning angle, theta.

For this problem, let f$($theta$)$ as $...$

f$($theta$)=$ e$^-$theta $*$ cos$(2$theta$)$

The distances at the initial positon $($theta $=0)$ and at the $90$ degree turn position

$($theta$=$ pi$/2\backslash$cong $1\mathrm{.}5708)$ are given as$...$

f$(0)=1$

f$(1\mathrm{.}5708)=0\mathrm{.}2079$

Since the sign of the function values at theta$=0,$ and theta$=1\mathrm{.}5708$ are different, there must exist a root of the function f$($theta$)$ between $[0,1\mathrm{.}5708].$

Using the False$-$Position Method, find the angle, theta, such that all four legs are on the surface.

Find the root estimate up to iteration, $,$ equals $2,$ and complete the table below with a four$-$decimal place.

Show all the calculations and steps to get answers.