Plug in the given values:
We need to identify two values, a and b, of h such that f(a)<0 and f(b)>0.
We can limit a and b by noting that 6h-h²≥0, that is, h(6-h)≥0, because the square root only applies to positive quantities. Therefore 0≤h≤6. The inverse cosine function also has a restricted argument which gives us the same range for h. Let a=0, b=6 and f(0)=-14 and f(6)=70.823 approx.
Applying the Regula-Falsi formula for the root:
f(0.9903)<0, so we know that the root lies between 0.9903 and 6. Readjust a to 0.9903 and leave b=6.
Next estimate for h:
f(1.3106)<0, so the root now lies between 1.3106 and 6. Readjust a to 1.3106 and leave b=6.
We continue this procedure until we reach a value of h which is stable to a sufficient number of decimal places and we end up with h=1.33096242.
Rounded to 4 decimal places this is h=1.3310.