A flagstaff stands on the top of a pole. The Angle if elevation of the pole and top of the flagstaff from the point on the ground is 30° and 60° respectively. If the height of the flagstaff is given by the minimum value of polynomial p(x) = 3x^2 - 15x + 22, then the height of the pole is
Answers
Solution
verified
15(
3
−1)m
tan45
∘
=
15
h
or h=15
tan60
∘
=
15
h+x
3
=
15
15+x
15
3
=15+x
15
3
−15=x
Hence, the length of flagstaff x=15(
3
−1) m
Given : A flagstaff stands on the top of a pole.
The Angle of elevation of the pole and top of the flagstaff from the point on the ground is 30° and 60° respectively.
The height of the flagstaff is given by the minimum value of polynomial p(x) = 3x² - 15x + 22,
To Find : the height of the pole
Solution:
p(x) = 3x² - 15x + 22
p'(x) = 6x - 15
p'(x) =0 => 6x - 15 = 0 => x = 5/2
p''(x) = 6 > 0
Hence p(x) is minimum at x = 5/2
=> p(x) = 3(5/2)² - 15(5/2) + 22
= 75/4 - 75/2 + 22
= 22 - 75/4
= 13/4
= 3.25
height of the pole = h
distance of observing point from base of the pole = d
tan 30° = h/d => 1/√3 = h/d => d = h√3
tan 60° = (h + 3.25)/d => √3 = (h + 3.25)/d
=> d = (h + 3.25)/√3
Equate d
(h + 3.25)/√3 = h√3
=> h + 3.25 = 3h
=> 2h = 3.25
=> h = 3.25/2
=> h = 1.625
Hence height of the pole is is 1.625
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