Answer: The distance to the lodge is 1514 ft
Step-by-step explanation: Please refer to the diagram attached.
The point where Alexa is standing at the top of the hill is marked as point A. The vertical height of the hill is from point A to point B which has been given as 1209 ft. Also, when Alexa looks down, she does that at an angle of depression measuring 53 degrees. Therefore while looking at the lodge, she forms an angle between her line of sight and the height of the hill. That angle is calculated as 90 degrees minus 53 degrees and that equals 37 degrees (as shown in the diagram). Note that her line of sight is AC since the ski lodge is at point C. Therefore the distance she must ski in order to get to point C (which shall be represented by X) shall be calculated using trigonometric ratios.
Note that we have a reference angle which is 37. We also have an adjacent which is 1209 (the side that lies between the reference angle and the right angle). And then we want to calculate the hypotenuse, yet unknown, (which is the side facing right angle). Since we have an adjacent, a hypotenuse and the reference angle, X can be calculated as
Cos A = Adjacent/Hypotenuse
Cos A = AB/AC
Cos 37 = 1209/X
By cross multiplication we now have X = 1209/Cos 37
By using our calculator or checking the table of values, Cos 37 = 0.7986
X = 1209/0.7986
X = 1513.899
Approximated to the nearest feet,
X = 1514 ft
Alexa have to walk about 911.05ft to reach the ski lodge.
Data;
To calculate the horizontal distance between the ski lodge and the hill, we can use trigonometric ratio SOHCAHTOA.
In this situation, we have the value of opposite and angle while we need to find the adjacent side to the angle.
Using the tangent of this angle, we can solve for this.
[tex]tan \theta = \frac{opposite}{adjacent} \\tan 53 = \frac{1209}{x}\\ x = \frac{1209}{tan 53}\\ x = 911.05ft[/tex]
From the calculation above, Alexa have to walk about 911.05ft to reach the ski lodge.
Learn more on trigonometric ratio here;
https://brainly.com/question/24349828
