Final answer:
To determine the height of the mountain, we can set up two trigonometric equations based on the angles of elevation from both lasers and solve for the height using the tangent function. This requires handling a system of two equations with two unknown variables.
Step-by-step explanation:
Two lasers are separated by 250 meters, and both are directed towards the top of a mountain. The angle of elevation of the first laser beam, which is closest to the mountain, is 40°. The angle of elevation for the second laser, which is further, is 35°. We can solve for the height of the mountain using trigonometric functions, specifically the tangent function which relates the angle of elevation to the opposite side (height of the mountain) and adjacent side (distance from the mountain for each laser).
Let's call the height of the mountain 'h'. For the first laser (closest to the mountain):
Tan(40°) = h / distance1
For the second laser (250 meters away from the first laser):
Tan(35°) = h / (distance1 + 250)
These two equations create a system that we can solve simultaneously to find the value of 'h'. We'll use a calculator to find the tangent values and then algebraically manipulate the equations to isolate 'distance1'. After finding the value of 'distance1', we'll substitute it back into either equation to solve for 'h', the height of the mountain.