Math in English

题目

Math in English
有人能帮我解一下这个题么?要解题不要翻译.
A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building.What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?(Give your answer correct to two decimal places.)

答案

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