let the angle between the ladder and ground be x.
And let the length of the ladder be y.
Then y = (8 cot (x) + 4)/(cos (x)) = 4 (2/(sin (x)) + 1/(cos (x)))
y` = -8 cot (x) csc (x) + 4 sec (x) tan (x)
1.let y` > 0 :
then - 8 cot (x) csc (x) + 4 sec (x) tan (x) > 0
so after simplify we have tan (x) > Power[2,(3)^-1]
(Attention :For all x value here,
the value of all trigonometrical function are positive)
so 0.90 < x < \[Pi]/2
2.let y` < 0
clearly we' ll get 0 < x < 0.90
so from case 1 and 2 we know that the length of ladder (y) will have \
the least value when x = 0.90
that is y (0.90) = 16.65 which will be length of the shortest ladder