Can you please help with this trig question:
The angle of elevation to the top of a nearby mountain is 46 degrees. After walking 1 km toward the mountain, you determine the angle of elevation to be 68 degrees. How high is the mountain?
Thank you for all your help.
as usual draw a diagram. If the height is x, then from distance x,
x/h = cotθ
so,
h cot 48° - h cot 68° = 1
Sure, I can help you with that!
To solve this problem, we can use trigonometry, specifically the tangent function.
Let's denote the height of the mountain as h.
From the information given, we can form a right-angled triangle with the horizontal distance between the observer and the mountain as the base, the height of the mountain as the height, and the line of sight from the observer to the top of the mountain as the hypotenuse.
Let's consider the first situation where the angle of elevation is 46 degrees. In this case, we have:
tan(46 degrees) = h / x
where x represents the horizontal distance between the observer and the mountain.
Now, let's consider the second situation where the angle of elevation is 68 degrees and the observer has walked 1 km toward the mountain. In this case, the horizontal distance between the observer and the mountain is x - 1. We have:
tan(68 degrees) = h / (x - 1)
Now we have two equations with two unknowns (h and x). We can solve this system of equations to find the value of h, which corresponds to the height of the mountain.
First, we can rearrange the first equation to solve for x:
x = h / tan(46 degrees)
Now, we substitute this expression for x in the second equation:
tan(68 degrees) = h / (h / tan(46 degrees) - 1)
Now, we can solve this equation to find the value of h, the height of the mountain.
Let me know if you would like to proceed with the calculations, or if you have any other questions!