Question

The angle of elevation of a ladder leaning against a wall is 60 degrees and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is ___ m. Type your answer below

Answer 1

Given :

• A ladder is leaning against a wall
• angle of elevation of ladder = 60°
• distance of foot of ladder from wall = 4.6 m

To find :

• Length of ladder

Figure :

$\setlength{\unitlength}{1cm}\begin{picture}(6,4)\thicklines\put(0,0){\line(1,0){4}}\put(0,0){\line(0,1){4}} \put(0,4){\line(1,-1){4}}\put(0,0.4){\line(1,0){0.4}}\put(0.4,0){\line(0,1){0.4}}\qbezier(3.2,0)(3,0.4)(3.5,0.5)\put(2.5,0.3){ 60^{\circ} }\put(-0.5,-0.5){A}\put(0,4.5){B}\put(4,-0.5){C}\put(1.5,-0.5){4.6\;m}\put(2.5,2.5){\bf{x}}\end{picture}$

• where AB is the wall
• AC is the distance of wall from foot of ladder i.e, AC = 4.6 m
• ∠ ACB is the angle of elevation of ladder i.e, ∠ ACB = 60°
• BC = x is the length of ladder

Solution :

as we know ,

$\implies\sf{cos\theta=\dfrac{adjacent\:side}{hypotenuse}}$

so,

$\implies\sf{cos\;60^{\circ}=\dfrac{4.6}{x}}$

putting value of cos 60° = 1 / 2

$\implies\sf{\dfrac{1}{2}=\dfrac{4.6}{x}}\\\\\\\implies\sf{x=4.6\times 2}\\\\\\\implies\boxed{\boxed{\sf{x=9.2\;\;m}}}$

Hence, Height of ladder is 9.2 m .

Answer 2

$\rule{300}3$

Given :-

• The angle of elevation of a ladder leaning against a wall = 60°.

• Distance of foot and ladder is 4.6 m.

To Find :-

• The length of the ladder = ?

Diagram :-

$\setlength{\unitlength}{1.6cm}\begin{picture}(6,2)\linethickness{0.5mm}\put(7.7,2.9){\large\sf{A}}\put(7.7,1){\large\sf{C}}\put(10.6,1){\large\sf{B}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\qbezier(10.5,1)(10,1.4)(8,2.9)\put(8.8,0.7){\sf{\large{4.6m}}}\put(7.4,2){\sf{\large{wall}}}\put(9.3,2){\sf{\large{ladder}}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\qbezier(9.8,1)(9.7,1.25)(10,1.4)\put(9.4,1.2){\sf\large{60^{\circ} }}\end{picture}$

• AB = ladder

• AC = wall

• $\angle$ B = 60°

• BC = 4.6 m

Solution :-

$\dashrightarrow\:$ $\sf cos \:\theta = \dfrac{Adjacent\:side}{Hypotenuse}$

$\:\:$

$\dashrightarrow\:$ $\sf cos\:60^{\circ} = \dfrac{4.6}{AB}$

$\:\:$

$\dashrightarrow\:$ $\sf \dfrac{1}{2} = \dfrac{4.6}{AB}$

$\:\:$

$\dashrightarrow\:$ $\sf AB = \dfrac{4.6 \times 2}{1}$

$\:\:$

$\dashrightarrow\:$ $\underline{\boxed{\gray{ \textbf{AB = 9.2 \:m}}}}$ $\red\bigstar$

$\:\:$

Therefore, The height of the ladder is 9.2 m.

$\rule{300}3$

$\gray\bigstar$ $\large{\underline{\purple{ \textbf{Extra\: Shots\::-}}}}$

$\dagger\:\sf Trigonometric\:Values :\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D \hat{e} fined\end{tabular}}$

$\rule{300}3$