Respuesta :
Answer:
- [tex]x = 5.65m [/tex]
- [tex]\theta = cos^{-1}\bigg(\dfrac{1}{3}\bigg)[/tex]
Step-by-step explanation:
Here it's given that a 6m ladder is leaned against a house, and the base of ladder is 2m away from the foot of the house. Here I am assuming x to be the height of the house.
Diagram :-
[tex] \begin{picture}(3,3)\setlength{\unitlength}{1cm} \linethickness{.6mm}\put(0,0){\line(1,0){4}}\put(0.01,0){\line(4,3){4}}\put(4,0){\line(0,1){3}}\put(4.3,1){$\bf x $} \put(2,-0.5){$\bf 2m$}\put(1,1.5){$\bf 6m$}\put(.9,0.2){$\bf\theta $}\put(2,4){$\boxed{\sf - RISH4BH }$}\end{picture} [/tex]
Using, Pythagoras theorem ,
[tex]\longrightarrow h^2=p^2+b^2 [/tex]
Substitute,
[tex]\longrightarrow (6m)^2=x^2+(2m)^2[/tex]
Simplify,
[tex]\longrightarrow 36m^2= x^2+4m^2 [/tex]
Subtracting 4m² on both sides,
[tex]\longrightarrow x^2 = 36m^2 -4m^2=32m^2 [/tex]
Put square root on both sides,
[tex]\longrightarrow x =\sqrt{32m^2}[/tex]
Simplify,
[tex]\longrightarrow \underline{\underline{ x = 5.65\ m}}[/tex]
Again in order to find out the angle of elevation,
[tex]\longrightarrow cos\theta =\dfrac{b}{h} [/tex]
Substitute,
[tex]\longrightarrow cos\theta =\dfrac{2m}{6m}[/tex]
Simplify,
[tex]\longrightarrow cos\theta =\dfrac{1}{3} [/tex]
Take cos inverse both sides,
[tex]\longrightarrow \underline{\underline{\theta =cos^{-1}\bigg(\dfrac{1}{3}\bigg)}}[/tex]
The length of the wall is 5.66m and the angle of elevation is 70.56°
Data;
- Length of Ladder (Hypothenuse) = x = 6m
- Distance from the wall to the foot of the ladder (adjacent)= y = 2m
- The length of the side of the house (opposite) = z
Pythagorean Theorem
To calculate the length of the wall or side of the house, we have to use Pythagorean theorem which is given as
[tex]x^2 = y^2 + z^2[/tex]
Let's substitute the values and solve for the length of the wall
[tex]x^2 = y^2 + z^2 \\6^2 = 2^2 + z^2 \\36 = 4 - z^2\\z^2 = 36 - 4\\z^2 = 32\\z = \sqrt{32} \\z = 5.66m[/tex]
The length of the wall is 5.66m.
Angle of Elevation
To calculate the angle of elevation, we can use trigonometric ratio SOHCAHTOA
since we have the value of all the sides, we can decide to use any of the ratios.
I will use sine angle to solve for the angle of elevation.
[tex]sin \theta = \frac{opposite}{hypothenuse} \\sin \theta = \frac{5.66}{6} \\sin \theta = 0.943\\\theta = sin^-^1 (0.943)\\\theta = 70.56^0[/tex]
The angle of elevation is 70.56°
Learn more on Pythagorean theorem and trigonometric ratio here;
https://brainly.com/question/6241673
