Respuesta :

Answer:

[tex]23.18 ft[/tex]

Step-by-step explanation:

To solve this problem, we make use of the geometry that forms between the wall, the floor and the ladder. A rectangular triangle is formed due to the right angle on the wall, see attached image.

We use the cosine function[tex]cos\theta=\frac{adjacentSide}{hypotenuse}=\frac{a}{h}[/tex]

Where [tex]\theta[/tex] is the angle, in this case is equal to: [tex]\theta =75[/tex].

[tex]a[/tex] is the adjacent side to the angle of 75: [tex]a=6[/tex], and [tex]h[/tex] is the hypotenuse of the triangle that is the length of the ladder, so in the last equation we clear for [tex]h[/tex]

[tex]h=\frac{a}{cos\theta}[/tex]

and substituting known values:

[tex]h=\frac{6}{cos(75)}[/tex]

[tex]h=\frac{6}{0.2588}[/tex]

[tex]h=23.18 ft[/tex]

The length of the ladder is [tex]23.18 ft[/tex]

Ver imagen rocioo

To solve the problem we must know the concept of trigonometric functions. The length of the ladder is 23.1822 ft.

What are the basic Trigonometric functions?

The basic trignometric fuctions are,

[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]

where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.

Given to us

The angle made by the ladder with the ground, θ = 75°

The foot of the ladder is 6 feet from the base of the wall.

As we know that the length between the ladder is 6 feet from the base of the wall, therefore, it is the base of the triangle, while the ladder makes 75° with the ground.

Substitute the values of the trigonometric function of Cos,

[tex]Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Cos \theta = \dfrac{\text{Distance between the base of the ladder and wall}}{\text{Length of the ladder}}\\\\\\Cos (75^o )=\dfrac{6}{\text{Length of the ladder}}\\\\\\Base = 23.1822\ ft[/tex]

Hence, the length of the ladder is 23.1822 ft.

Learn more about Trigonometric functions:

https://brainly.com/question/6904750

Otras preguntas