Question

two objects on horizontal plane in the same line of the foot of the cliff form with the top of cliff angles of elevation of 30° and45°. if the height of the cliff is100m calculate the distance between the two object​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

Height of the cliff be AB with base B and top A.

Let two objects are at C and D on the same side of cliff.

Angle of elevation from point C to the top of cliff is 30° and from point D is 45°.

Let assume that distance between C and D be 'x' m. and distance between A and D be 'y' m.

Height of the cliff, AB = 100 m

Now, Consider

$\boxed{ \bf{ \: In \: \triangle \: ABD}}$

$\rm :\longmapsto\:tan45 \degree \: = \: \dfrac{AB}{BD}$

$\rm :\longmapsto\:1 = \dfrac{100}{y}$

$\bf\implies \:y = 100 \: m$

Now, Consider

$\boxed{ \bf{ \: In \: \triangle \: ABC}}$

$\rm :\longmapsto\:tan30 \degree \: = \: \dfrac{AB}{BC}$

$\rm :\longmapsto\:\dfrac{1}{ \sqrt{3} } = \dfrac{100}{x + y}$

$\rm :\longmapsto\:x + y = 100 \sqrt{3}$

$\rm :\longmapsto\:x + 100 = 100 \sqrt{3}$

$\red{\bigg \{ \because \: y = 100 \: \bigg \}}$

$\rm :\longmapsto\:x = 100 \sqrt{3} - 100$

$\rm :\longmapsto\:x = 100( \sqrt{3} - 1)$

$\rm :\longmapsto\:x = 100( 1.732 - 1)$

$\rm :\longmapsto\:x = 100( 0.732)$

$\bf\implies \:x = 73.2 \: m$

So, Distance between the objects is 73.2 m.

Additional Information :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$