a) 67°
b) 2.6 ft
Step-by-step explanation:
Part (a)
This problem can be modeled as a right triangle (see attachment 1), where the wall and the ground are the legs, and the ladder is the hypotenuse. Therefore, we can use the sine trig ratio to find the angle.
Trig Ratio
[tex]\sf \sin(\theta)=\dfrac{O}{H}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- H is the hypotenuse (the side opposite the right angle)
Given:
- [tex]\theta[/tex] = x
- O = wall = 9.2
- H = ladder = 10
Substituting the given values into the ratio and solving for x:
[tex]\implies \sf \sin(x)=\dfrac{9.2}{10}[/tex]
[tex]\implies \sf x=\sin^{-1}\left(\dfrac{9.2}{10}\right)[/tex]
[tex]\implies \sf x=67^{\circ}\:(nearest\:degree)[/tex]
Part (b)
(see attachment 2)
Let y be the distance the foot of the ladder and the foot of the building
We can find y by using the cos trig ratio:
[tex]\sf \cos(\theta)=\dfrac{A}{H}[/tex]
where:
- [tex]\theta[/tex] is the angle
- A is the side adjacent the angle
- H is the hypotenuse (the side opposite the right angle)
Given:
- [tex]\theta[/tex] = 75°
- A = grounds = y
- H = ladder = 10
Substituting the given values into the ratio and solving for x:
[tex]\implies \sf \cos(75^{\circ})=\dfrac{y}{10}[/tex]
[tex]\implies \sf y=10\cos(75^{\circ})[/tex]
[tex]\implies \sf y=2.6\:ft\:(nearest\:tenth)[/tex]