A road runs from the base of a mountain. From two points 235 meters apart on the road, the angles of elevation to the top of the mountain are 43 and 30. how high above the road is the mountaintop?
Draw a diagram. If the nearer point is a distance x from the base of the mountain, then the height h can be figured from
h/x = tan 43°
h/(x+235) = tan 30°
x = h/tan 43°
x = h/tan 30° - 235
h/tan 43° = h/tan 30° - 235
h = 235/(cot30° - cot43°)
h = 356.2 m
not much of a mountain.
To find the height of the mountaintop, we need to use trigonometry. Let's break down the problem step by step.
Step 1: Draw a diagram:
Draw a triangle where the base represents the road, one side represents the height of the mountain, and the other side connects the base to the mountaintop.
Step 2: Label the diagram:
Label the base of the triangle as "b," the height of the mountain as "h," and the distance between the two points as "d" (given as 235 meters).
Step 3: Determine the trigonometric ratios:
In this case, we need to use the tangent function because we are given the angle of elevation.
tan(angle) = opposite / adjacent
For the first angle of elevation (43 degrees):
tan(43) = h / b
For the second angle of elevation (30 degrees):
tan(30) = h / (b + d)
Step 4: Solve the equations:
We have two equations with two unknowns (h and b). We can solve the equations simultaneously to find the values.
From the first equation:
h = b * tan(43)
Substitute this value of h in the second equation:
tan(30) = (b * tan(43)) / (b + d)
Now we can solve this equation to find the value of b.
Step 5: Substitute the value of b to find h:
Once we have the value of b, we can substitute it back into the equation from step 2:
h = b * tan(43)
Solve this equation to find the final answer, which will be the height of the mountaintop above the road.