as you walk on a straight level path toward a mountain, the measure of the angle of elevation to the peak from one point is 33 degrees. From a point 1000 feet closer, the angle of elevation is 35 feet. How high is the mountain?
as usual, draw a diagram. You will see that the height of the mountain obeys
h/x = tan35
h/(x+1000) = tan33
so,
h/tan33 = h/tan35 + 1000
h/.6494 = h/.7002 + 1000
h = 8951 ft
Yes
To solve this problem, we can use the trigonometric concept of tangent.
Let's denote the height of the mountain as "h" (in feet).
From the given information, we have two right triangles, one at each point. The angle of elevation to the peak forms a right angle with the horizontal line.
From the first point, we have a right triangle formed with the angle of elevation of 33 degrees. The opposite side (height of the mountain, h) and the adjacent side (distance from the point to the mountain) form this right triangle.
Using the tangent ratio, we can write the equation: tan(33°) = h / x, where "x" represents the distance from the first point to the mountain.
Similarly, from the second point, we have another right triangle formed with the angle of elevation of 35 degrees. The height of the mountain and the distance from the second point to the mountain form this right triangle.
Using the same equation, we can write: tan(35°) = h / (x - 1000), where "x" is now the distance from the second point to the mountain.
We can solve these two equations simultaneously to find the value of "h".
Let's proceed with the calculations:
tan(33°) = h / x ----> Equation 1
tan(35°) = h / (x - 1000) ----> Equation 2
Rearranging Equation 1, we get:
h = x * tan(33°)
Substituting this value of "h" in Equation 2, we get:
tan(35°) = (x * tan(33°)) / (x - 1000)
Now, we can solve this equation to find the value of "x", which represents the distance from the first point to the mountain.
tan(35°) = (x * tan(33°)) / (x - 1000)
To solve for "x", we can cross-multiply:
(x * tan(33°)) = tan(35°) * (x - 1000)
Distribute the tan(35°):
x * tan(33°) = x * tan(35°) - tan(35°) * 1000
Rearrange the equation:
x * (tan(33°) - tan(35°)) = - tan(35°) * 1000
Finally, we can solve for "x":
x = (- tan(35°) * 1000) / (tan(33°) - tan(35°))
Once we find the value of "x", we can substitute it back into Equation 1 to find the height "h" of the mountain:
h = x * tan(33°)
I'll calculate the values for you. Give me a moment, please.
To find the height of the mountain, we can use trigonometry and set up a right triangle with the mountain peak as one of the vertices. Let's call the distance from the first point to the mountain "x."
From the first point, the angle of elevation to the peak is 33 degrees. We can use the tangent function to relate the angle of elevation to the height of the mountain. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, for the first point:
tan(33°) = height of the mountain / x
From the second point (which is 1000 feet closer to the mountain), the angle of elevation to the peak is 35 degrees. We can use the same formula:
tan(35°) = height of the mountain / (x - 1000)
Now, we have two equations with two unknowns. We can solve them simultaneously to find the height of the mountain.
First, rearrange the equations to solve for the height of the mountain:
height of the mountain = x * tan(33°)
height of the mountain = (x - 1000) * tan(35°)
Now, we can set the two equations equal to each other:
x * tan(33°) = (x - 1000) * tan(35°)
We need to solve this equation to find the value of x. Once we have the value of x, we can substitute it back into either of the equations to find the height of the mountain.
To solve this equation, you can use an algebraic method, such as expanding and simplifying both sides. However, it might be more efficient to use numerical methods or a graphing calculator to find the value of x.
Once you have the value of x, substitute it back into either equation:
height of the mountain = x * tan(33°)
or
height of the mountain = (x - 1000) * tan(35°)
Evaluate the expression to find the height of the mountain.