In this problem, we need to calculate the altitude $h$ of the pilot's plane. The setup involves two angles of depression (27° and 53°) and the horizontal distance of 9 miles between two mileposts on the highway.

To solve this, we can use trigonometry, specifically the tangent function, because we know the angles of depression and the horizontal distances:

Step 1: Label the problem

Let’s label the problem as follows:

• $A$: The position of the airplane.
• $B$: The first milepost (closer to the plane, with an angle of depression of 53°).
• $C$: The second milepost (farther from the plane, with an angle of depression of 27°).

The horizontal distance between the mileposts is $BC = 9$ miles.

Step 2: Set up equations using the tangent function

The altitude $h$ of the plane and the horizontal distances from the plane to the mileposts are related by the tangent of the angles of depression.

• For the first milepost (closer to the plane, at point $B$): $\tan(53^\circ) = \frac{h}{x}$ where $x$ is the horizontal distance from the plane to the first milepost.

  Rearranging the equation for $h$, we get: $h = x \cdot \tan(53^\circ)$

• For the second milepost (farther from the plane, at point $C$): $\tan(27^\circ) = \frac{h}{x + 9}$ where $x + 9$ is the total horizontal distance from the plane to the second milepost.

  Rearranging this equation for $h$, we get: $h = (x + 9) \cdot \tan(27^\circ)$

Step 3: Solve the system of equations

Since both expressions represent $h$, we can set them equal to each other: $x \cdot \tan(53^\circ) = (x + 9) \cdot \tan(27^\circ)$

Now we can solve for $x$. Let’s do the calculation:

$x \cdot 1.327 = (x + 9) \cdot 0.5095$

Now distribute on the right-hand side: $x \cdot 1.327 = x \cdot 0.5095 + 4.5855$

Subtract $x \cdot 0.5095$ from both sides: $x \cdot (1.327 - 0.5095) = 4.5855$ $x \cdot 0.8175 = 4.5855$

Now solve for $x$: $x = \frac{4.5855}{0.8175} \approx 5.61 \text{ miles}$

Step 4: Find the altitude $h$

Now that we know $x \approx 5.61$ miles, we can use one of the original tangent equations to find the altitude. Using the first equation:

$h = x \cdot \tan(53^\circ) = 5.61 \cdot 1.327 \approx 7.44 \text{ miles}$

Thus, the altitude of the plane is approximately 7.44 miles.

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Would you like further details or have any questions?
Here are some related questions to explore this topic:

1. How would the solution change if the distance between mileposts were different?
2. How does the altitude relate to different angles of depression?
3. What other trigonometric functions could be used in such problems?
4. How could this problem be solved if the altitude were known but the angles were unknown?
5. How could this approach be adapted if the highway were not straight?

Tip: Always make sure to label your diagrams clearly in trigonometric problems. It helps in setting up equations correctly.