A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.4 km apart, to be 35 degrees and 47 degrees , as shown in the figure.
no figure, but if they are both ahead of him, then
his height h (in km) can be determined by
h cot35° - h cot47° = 6.4
To solve this problem, we can use trigonometry.
Let's refer to the diagram and label the given information:
- Angle of depression to the first milepost: 35 degrees
- Angle of depression to the second milepost: 47 degrees
- Distance between the two mileposts: 6.4 km
Now, we can break down the problem into two separate triangles and solve for the height of the pilot.
In the first triangle, let's consider the angle of depression of 35 degrees and label the opposite side 'h'.
Step 1: Calculate the height of the pilot to the first milepost:
We can use the tangent function to find the height (h):
tan(35) = h / 6.4 km
Rearranging the equation, we get:
h = 6.4 km * tan(35)
Calculating h:
h ≈ 5.70 km
Therefore, the height of the pilot to the first milepost is approximately 5.70 km.
Moving on to the second triangle, let's consider the angle of depression of 47 degrees and label the opposite side 'h2'.
Step 2: Calculate the height of the second milepost:
Again, we can use the tangent function:
tan(47) = h2 / 6.4 km
Rearranging the equation, we get:
h2 = 6.4 km * tan(47)
Calculating h2:
h2 ≈ 7.15 km
Therefore, the height of the pilot to the second milepost is approximately 7.15 km.
To summarize:
- The height of the pilot to the first milepost is approximately 5.70 km.
- The height of the pilot to the second milepost is approximately 7.15 km.
To solve this problem, we can create a diagram to visualize the scenario. Let's label the points as follows:
A: Location of the pilot
B: First milepost
C: Second milepost
Next, we can label the distances as follows:
AB = x
BC = 6.4 km
Now, we can use trigonometric ratios to find the value of x.
From the diagram and the given information, we can observe that:
∠BAC = 180° - 35° = 145°
∠CAB = 180° - 47° = 133°
To find the length of AB, we can use the tangent ratio, given by:
tan(∠CAB) = AB / BC
Plugging in the values, we have:
tan(133°) = x / 6.4 km
Now, we can solve for x by rearranging the equation:
x = 6.4 km * tan(133°)
Using a calculator, we can find the value of tan(133°) to be approximately 3.799.
Substituting this value, we get:
x ≈ 6.4 km * 3.799 ≈ 24.276 km
Therefore, the distance AB, or the distance between the pilot and the first milepost, is approximately 24.276 km.