A-man-observes-that-when-he-moves-up-a-distance-c-metres-on-a-slope-the-angle-of-depression-of-a-point-on-the-horizontal-plane-from-the-base-of-the-slope-is-30-and-when-he-moves-up-further-a-distan

Question Number 15393 by Tinkutara last updated on 10/Jun/17

A man observes that when he moves up

a distance c metres on a slope, the

angle of depression of a point on the

horizontal plane from the base of the

slope is 30 °, and when he moves up

further a distance c metres, the angle of

depression of that point is 45 ° . The

angle of inclination of the slope with the

horizontal is ?

Answered by mrW1 last updated on 10/Jun/17

let

a = distance of the point to foot of slope

α = angle of slope

γ = \frac{a}{c}

(a + c \cos α) \tan 30 ° = c \sin α

(γ + \cos α) \frac{1}{\sqrt{3}} = \sin α \dots (i)

(a + 2 c \cos α) \tan 45 ° = 2 c \sin α

(γ + 2 \cos α) = 2 \sin α \dots (ii)

(ii) / (i) :

\frac{γ + 2 \cos α}{γ + \cos α} \times \sqrt{3} = 2

γ \sqrt{3} + 2 \sqrt{3} \cos α = 2 γ + 2 \cos α

2 (\sqrt{3} - 1) \cos α = (2 - \sqrt{3}) γ

2 (\sqrt{3} - 1) \cos α = 2 (2 - \sqrt{3}) (\sin α - \cos α)

\cos α = (2 - \sqrt{3}) \sin α

\tan α = \frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3}

α = \tan^{- 1} (2 + \sqrt{3}) = 75 °

Commented by mrW1 last updated on 10/Jun/17

Commented by Tinkutara last updated on 10/Jun/17

Thanks Sir!