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?
To find the height of the mountaintop, we can use the concept of trigonometry and the properties of right triangles.
Let's denote the height of the mountaintop as 'h', and the distance from the base of the mountain to the point on the road with the angle of elevation of 43 degrees as 'x'. Therefore, the distance from the base of the mountain to the point on the road with the angle of elevation of 30 degrees will be '235 - x'.
Using the tangent function, we can set up the following equations:
tan(43) = h / x (Equation 1)
tan(30) = h / (235 - x) (Equation 2)
Now, let's solve the equations simultaneously:
From Equation 1, we can rearrange it to solve for h:
h = x * tan(43)
Substituting this value of h in Equation 2:
tan(30) = (x * tan(43)) / (235 - x)
Now, we have a single equation with a single variable, x. We can solve for x by manipulating the equation further:
tan(30) * (235 - x) = x * tan(43)
Using the values of tan(30) ≈ 0.577 and tan(43) ≈ 0.931, we can simplify the equation:
0.577 * (235 - x) = 0.931 * x
Solving for x:
135.395 - 0.577x = 0.931x
Combining like terms:
1.508x = 135.395
Dividing both sides by 1.508:
x ≈ 89.84
Now, substituting the value of x back into Equation 1:
h = x * tan(43)
≈ 89.84 * 0.931
≈ 83.66
Therefore, the height of the mountaintop is approximately 83.66 meters above the road.