Bisection Method of Solving a Nonlinear Equation – More Examples ...

03.03.2 Chapter 03.03
Solution
From the physics of the problem, the dipstick would be wet between h 0 and h 2r ,
where
r radius of the tank,
that is
0 h 2r
0 h 2(3)
0 h 6
Let us assume
h
0,
hu
6
Check if the function changes sign between h and h .
3 2
0
0 90 3.8197 3. 8197
f ( h
) f
3 2
f(h u
) f (6) (6) 9(6) 3.8197 104.18
Hence
f h
f
hu
f 0f
6
3.8197
104.18 0
So there is at least one root between h and h that is between 0 and 6.
u
Iteration 1
The estimate of the root is
h hu
hm
2
0 6
2
3
3 2
f hm
f 3 3 9 3 3.1897 50.
180
f h
f
hm
f 0f
3
3.1897
50.180 0
Hence the root is bracketed between h
and hm
, that is, between 0 and 3. So, the lower and
upper limits of the new bracket are
h
0,
hu
3
At this point, the absolute relative approximate error
a
cannot be calculated, as we do not
have a previous approximation.
Iteration 2
The estimate of the root is
h hu
hm
2
0 3
2
1.5
u