Question

αɳรωεɾ ƭɦเร φµεรƭเσɳ ƒɾɳ∂ร...

รραɱ αɳรωεɾร ωเℓℓ ɓε ɾερσɾƭε∂​

Answer 1

$\huge\sf\color{lime}{Figure}$

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\linethickness{0.4mm}\put(8,1){\line(1,0){3}}\put(8,1){\line(0,2){2}}\put(11,1){\line(0,3){4.4}}\put(8,3){\line(3,0){3}}\put(11.1,2){\sf50 m}\put(7.3,2){\sf50 m}\put(11.1,4){\sf x m}\qbezier(8,1)(10.7,5)(11,5.4)\qbezier(8,3)(10.5,5)(11,5.4)\qbezier(8.65,1)(8.7,1.4)(8.4,1.6)\put(8.7,1.2){\sf60^{\circ} }\qbezier(8.5,3.4)(8.8,3.2)(8.7,3)\put(8.8,3.2){\sf45^{\circ} }\put(7.7,3){\large{Y}}\put(7.7,1){\large{X}}\put(11.1,1){\large{P}}\put(11.1,3){\large{R}}\put(11.1,5.3){\large{Q}}\put(9.5,0.8){\sf y m}\end{picture}$

$\huge\sf\pink{Answer}$

☞ Distance inbetween = 68.25 m

☞ Height of tower = 118.25 m

$\rule{110}1$

$\huge\sf\blue{Given}$

✭ The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively.

➢ A Building ( YX) of height 50 m

➢ A Tower ( QP) of height (x + 50) m

➢ Horizontal Distance between Tower and Building be XP i.e. y

$\rule{110}1$

$\huge\sf\gray{To\:Find}$

☆ Height of the tower

☆ Horizontal distance between the tower and the building

$\rule{110}1$

$\huge\sf\purple{Steps}$

❍ $\underline{\textsf{In \triangle QRY, where \measuredangle R is 90 ^{\circ} :}}$

$\twoheadrightarrow\sf \tan(\theta)=\dfrac{Perpendicular}{Base}\\ \\ \twoheadrightarrow\sf \tan(45^{\circ})=\dfrac{QR}{YR}\\\\ \twoheadrightarrow\sf \tan(45^{\circ})=\dfrac{QR}{XP}\qquad[\because\:YR=XP=y]\\\\ \twoheadrightarrow\sf 1 = \dfrac{x}{y}\\\\ \color{aqua}{\twoheadrightarrow\sf y = x \qquad...\:eq \:(l)}$

❍ $\underline{\textsf{In \triangle QPX, where \measuredangle P is 90 ^{\circ} :}}$

$\longmapsto\sf \tan(\theta)=\dfrac{Perpendicular}{Base}\\\\ \longmapsto\sf \tan(60^{\circ})=\dfrac{QP}{XP}\\ \\ \longmapsto\sf \sqrt{3} = \dfrac{x + 50}{y}\\\\ \longmapsto\sf \sqrt{3} = \dfrac{x + 50}{x}\qquad[\because\:y=x\:from\:eq.\:l]\\\\ \longmapsto\sf \sqrt{3}x = x + 50\\\\ \longmapsto\sf \sqrt{3}x - x = 50\\\\ \longmapsto\sf x( \sqrt{3} - 1) = 50\\\\ \longmapsto\sf x = \dfrac{50}{\sqrt{3} - 1}\\\\ \longmapsto\sf x =\dfrac{50}{\sqrt{3} - 1} \times\dfrac{ \sqrt{3} + 1 }{\sqrt{3} + 1}\\\\ \longmapsto\sf x = \dfrac{50( \sqrt{3} + 1)}{3 - 1}\\\\ \longmapsto\sf x = \dfrac{50(1.73 + 1)}{2}\\\\ \longmapsto\sf x = 25 \times 2.73\\\\ \color{pink}{\longmapsto\sf x = 68.25\:m = y} \qquad \bigg\lgroup\sf Distance\:in\:Between\bigg\rgroup$

❍ $\underline{\textsf{Height of Tower :}}$

$\dashrightarrow\sf\:\:Tower=QP\\\\ \dashrightarrow\sf\:\:Tower=x+50\\\\ \dashrightarrow\sf\:\:Tower=68.25\:m+50\\\\\orange{\dashrightarrow\sf\:\:Tower=118.25\:m}\qquad \bigg\lgroup\sf Height\:of\:Tower\bigg\rgroup$

$\rule{170}3$