In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive x = 9 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain. (Round your answer to one decimal place.)
the picture in the link below:
www.webassign.net/larprecalcaga5/5-2-080.gif
Tan3.5 = h/d, h = d*Tan3.5.
Tan9 = h/(d-9), h = (d-9)*Tan9.
h = d*Tan3.5 = (d-9)*Tan9.
d*Tan3.5 = (d-9)*Tan9, Divide both sides by Tan3.5:
d = (d-9)*2.59,
d = 2.59d - 23.3,
-1.59d = -23.3,
d = 14.7 miles.
h = d*Tan3.5 = 14.7*Tan3.5 = 0.90 miles.
To solve this problem, we can use trigonometry and create a right triangle with the base being the distance we drive towards the mountain (x), the height being the height of the mountain, and the angle of elevation being the angle from the ground to the peak of the mountain.
Let's consider the first scenario when the angle of elevation is 3.5°. We can use the tangent function to relate the height of the mountain to the distance we are from it:
tan(angle) = opposite / adjacent
tan(3.5°) = height / x
Simplifying the equation, we get:
height = x * tan(3.5°)
Now let's consider the second scenario when the angle of elevation is 9°. Again, using the tangent function, we can relate the height of the mountain to the new distance we are from it:
tan(9°) = height / (x - 9)
Simplifying the equation, we get:
height = (x - 9) * tan(9°)
Now we can set these two expressions for height equal to each other:
x * tan(3.5°) = (x - 9) * tan(9°)
To solve for the height, we need to find the value of x. Rearranging the equation, we have:
tan(3.5°) / tan(9°) = (x - 9) / x
Now we can solve for x:
x = (tan(3.5°) / tan(9°)) / (1 - tan(3.5°) / tan(9°))
Let's calculate x using this equation:
x = (tan(3.5°) / tan(9°)) / (1 - tan(3.5°) / tan(9°))
x ≈ 42.6 miles
Now that we have the value of x, we can substitute it into either of the earlier equations to find the height. Let's use the first equation:
height = x * tan(3.5°)
height ≈ 42.6 * tan(3.5°)
height ≈ 2.8 miles
Therefore, the approximate height of the mountain is 2.8 miles.
To approximate the height of the mountain, we can use trigonometry. In this case, we will use the tangent function. Let's call the height of the mountain "H".
First, let's analyze the diagram. We have two right triangles: one before driving closer to the mountain and one after driving closer.
In the first triangle:
- The opposite side is the height of the mountain, H.
- The adjacent side is the distance you are from the mountain, x (which is unknown).
- The angle between the adjacent and hypotenuse sides is 3.5°.
In the second triangle:
- The opposite side is now H.
- The adjacent side is x + 9 miles.
- The angle between the adjacent and hypotenuse sides is 9°.
Now, we can use the tangent function to set up two equations:
In the first triangle:
tan(3.5°) = H / x
In the second triangle:
tan(9°) = H / (x + 9)
To find the height of the mountain, we need to solve these two equations simultaneously.
1. Solve the first equation for x:
x = H / tan(3.5°)
2. Substitute this expression for x into the second equation:
tan(9°) = H / (H / tan(3.5°) + 9)
3. Simplify the second equation by multiplying both sides by (H / tan(3.5°) + 9):
tan(9°) * (H / tan(3.5°) + 9) = H
4. Distribute the left side of the equation:
(H * tan(9°)) / tan(3.5°) + (9 * tan(9°)) = H
5. Group the terms with H on one side:
(H * tan(9°)) / tan(3.5°) - H = -(9 * tan(9°))
6. Factor out H:
H * ((tan(9°) / tan(3.5°)) - 1) = -(9 * tan(9°))
7. Divide both sides by ((tan(9°) / tan(3.5°)) - 1):
H = -(9 * tan(9°)) / ((tan(9°) / tan(3.5°)) - 1)
Now, we can plug the values into a calculator to approximate the height of the mountain using the above equation. Round your answer to one decimal place to match the problem's instructions.
You can find all the angles of the obtuse triangle, and you have one side.
So, find the side which is also the hypotenuse of the other triangle using the sine law.
Now in the right-angled triangle, use sin 9 = height/hypotenuse to find the height