Hey guys this is a precalculus honors question and im stuck, appreciate it if i could get your help.
Lauri is a sea level and uses surveying equipment to sight the top of Mount Tam. The angle of elevation to the top is
22 degrees 10 minutes. She moves 1000 ft toward Mount Tan and finds that the angle of elevation to the top is
32 degrees 40 minutes. What is the height of Mount Tam?
This involves adding vectors. My teacher showed us the formula for how to add vectors, but I'm confused on how to set it up so i can add the two vectors. And no im not asking you guys to help me do my homework, but if you can just show me how to do this one, maybe ill understand how to solve the rest of them. :D
Also, can someone explain to me how to find the direction (the degree of angle), because i know that its different when there is a "from" in the question, implying a negative vector.
To solve this problem, you can use trigonometry to set up two right-angled triangles: one at the original position and another at the new position.
Let's establish some notation:
- Let d be the distance between Lauri's original position and Mount Tam.
- Let h be the height of Mount Tam.
First, let's set up the triangle at the original position. Since Lauri is at sea level, the opposite side is just the height of Mount Tam (h), and the adjacent side is the distance between Lauri and Mount Tam (d).
Using the given angle of elevation at the original position (22 degrees 10 minutes), convert it to decimal degrees by adding the minutes component (10/60) to the degrees component:
Angle of elevation (in decimal degrees) = 22 + (10/60) = 22.167 degrees
Now, using trigonometric ratios, we can set up an equation with respect to the original position triangle:
tan(22.167) = h/d
This equation relates the height of Mount Tam and the distance between Lauri and Mount Tam at the original position.
Next, let's set up the triangle at the new position. Lauri moves 1000 ft closer to Mount Tam, so the distance between Lauri and Mount Tam becomes (d - 1000). The angle of elevation at the new position is given as 32 degrees 40 minutes. Convert it to decimal degrees as before:
Angle of elevation (in decimal degrees) = 32 + (40/60) = 32.667 degrees
Again, using trigonometric ratios, we can set up another equation with respect to the new position triangle:
tan(32.667) = h/(d - 1000)
This equation relates the height of Mount Tam and the new distance between Lauri and Mount Tam after moving closer.
Now, we have a system of two equations with two variables (h and d):
Equation 1: tan(22.167) = h/d
Equation 2: tan(32.667) = h/(d - 1000)
To solve this system, you can use algebraic methods such as substitution or elimination. Once you find the values of h and d, you can substitute them back into either equation to find the height of Mount Tam (h).
I hope this explanation helps you understand how to set up and solve problems involving adding vectors using trigonometry!