Question

The angles of depression of the top and bottom of 12 m tall building from the top of a multistoried
building are 30° and 45° respectively. Find the height of the multi-storeyed building and the
distance between the two buildings.

Answer 1

Answer:

$\:\boxed{\begin{aligned}& \:\sf \:Height\:of\:multi-storeyed\:building = 6(3 + \sqrt{3}) \: m \: \\ \\& \:\sf \: Distance\:between\:two\:buildings=6(3 + \sqrt{3}) \: m \end{aligned}} \qquad \:$

Step-by-step explanation:

Let assume that AB represents the building and CD represents the multi-storeyed building such that AB = 12 m.

Now, from top C of the multi-storeyed building, the angle of depression of top and bottom of building AB be 30° and 45° respectively.

Let assume that CD = h m and BD = x m.

From A, draw AE perpendicular to CD intersecting CD at E.

Now, AE = BD = x m

So, CE = CD - DE = CD - AB = (h - 12) m

Now, In right-angle triangle CDB

$\sf \: tan45 \degree \: = \: \dfrac{CD}{DB}$

$\sf \: 1 \: = \: \dfrac{h}{x}$

$\sf\implies \sf \: h = x$

Now, In right-angle triangle CEA

$\sf \: tan30 \degree \: = \: \dfrac{CE}{EA}$

$\sf \: \dfrac{1}{\sqrt{3}} \: = \: \dfrac{h - 8}{x}$

$\sf \: \dfrac{1}{ \sqrt{3} } \: = \: \dfrac{h - 8}{h}$

$\sf \: \sqrt{3}(h - 12) \: = \: h$

$\sf \: \sqrt{3}h - 12 \sqrt{3} \: = \: h$

$\sf \: \sqrt{3}h - h = 12 \sqrt{3}$

$\sf \: (\sqrt{3} -1) h = 12 \sqrt{3}$

$\sf \: h = \dfrac{12 \sqrt{3} }{ \sqrt{3} - 1 }$

On rationalizing the denominator, we get

$\sf \: h = \dfrac{12\sqrt{3} }{ \sqrt{3} - 1 } \times \dfrac{ \sqrt{3} + 1}{ \sqrt{3} + 1 }$

$\sf \: h = \dfrac{12 \sqrt{3} ( \sqrt{3} + 1)}{( \sqrt{3})^{2} - {(1)}^{2} }$

$\sf \: h = \dfrac{12\sqrt{3} ( \sqrt{3} + 1)}{3 - 1 }$

$\sf \: h = \dfrac{12 \sqrt{3} ( \sqrt{3} + 1)}{2}$

$\sf \: h = 6 \sqrt{3} ( \sqrt{3} + 1)$

$\sf\implies \sf \: h = 6(3 + \sqrt{3}) \: m$

So,

$\sf\implies \sf \: x = 6 (3 + \sqrt{3}) \: m$

Hence,

$\:\boxed{\begin{aligned}& \:\sf \:Height\:of\:multi-storeyed\:building = 6(3 + \sqrt{3}) \: m \: \\ \\& \:\sf \: Distance\:between\:two\:buildings=6(3 + \sqrt{3}) \: m \end{aligned}} \qquad \:$