Question

Two walls of a canyon form the walls of a steady flowing river. From a point on the shorter wall, the angle of elevation to the top of the opposing wall is 20° and the angle of depression to the bottom of the opposing wall is 230 feet. Using the appropriate right triangle solving strategies, solve for the following: (Do not round intermediate calculated values. Only the final answer should be rounded to one decimal place.)

the height of the short wall (x)
the height of the tall wall (y)
the distance between the canyon walls (z)

How do I solve this and get to the answer.

Answer 1

We found that the height of the short wall "x" is 162.6 ft, the height of the tall wall "y" is 221.8 ft, and the distance between the canyon walls "z" is 162.6 ft.

To find the x, y, and z values we need to denote the right triangles from top to bottom as triangles 1, 2, and 3.

1. Finding the height of the short wall "x"

We can find the height of the short wall "x" (in triangle 3) with the following trigonometric function:

$cos(\theta) = \frac{x}{H}$

Where:

H: is the hypotenuse = 230 ft

θ: is the angle between x and H.

Knowing that the sum of θ and the angle 45° must be equal to 90°, θ is:

$\theta = 90 - 45 = 45$

Hence, the height of the short wall "x" is:

$x = cos(\theta)*H = 230cos(45) = 162.6 ft$                

2. Finding the height of the tall wall "y"  

The height of the tall wall "y" is given by the sum of the bases of the two first right triangles (the right triangles 1 and 2):

$y = y_{1} + y_{2}$        

Where y₁ and y₂ can be calculated with the tangent and sine trigonometric functions.  

$y_{1} = A*tan(20)$        

$y_{2} = 230sin(45)$

Where A is the adjacent side to the angle 20°.

$y = A*tan(20) + 230sin(45)$      

Since the right triangles 2 and 3 form a square, with all the sides equals to x, we have:        

$A = z = y_{2} = x = 230cos(45)$        

We can use 230cos(45) or 230sin(45) to calculate y₂, so the height of the tall wall "y" is:      

$y = y_{1} + y_{2} = A*tan(20) + 230cos(45) = 230cos(45)tan(20) + 230cos(45) = 221.8 ft$                        

3. Finding the distance between the canyon walls "z"

As we said above, the "z" value is the same as "x", then:

$z = x = 230cos(45) = 162.6 ft$

Learn more about trigonometric functions here: https://brainly.com/question/14272510?referrer=searchResults

I hope it helps you!