A plane is flying at an altitude of 12,000 m. From the pilot, the angle of depression to the airport tower is 32°. How far is the tower from a point directly beneath the plane? Round to the nearest whole number 32° :12,000 m < < X (1 point) meters

Answer 1

To find the horizontal distance from the point directly beneath the plane to the airport tower, we can use the tangent of the angle of depression. The angle of depression from the pilot to the tower is 32°, and the altitude of the plane is 12,000 m.

In right triangle trigonometry, the tangent of an angle is given by:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Where:

• The "opposite" side is the altitude of the plane (12,000 m).
• The "adjacent" side is the horizontal distance from the point directly beneath the plane to the tower (let's call this \(X\)).

From the problem, we can write:

\[ \tan(32°) = \frac{12,000}{X} \]

Now, we can rearrange this to solve for \(X\):

\[ X = \frac{12,000}{\tan(32°)} \]

Next, we calculate \(\tan(32°)\):

\[ \tan(32°) \approx 0.6249 \]

Now, plugging this value into the equation for \(X\):

\[ X = \frac{12,000}{0.6249} \approx 19,207.6 \]

Rounding to the nearest whole number, the tower is approximately:

\[ \boxed{19208} \text{ meters} \]

So, the distance from the point directly beneath the plane to the airport tower is about 19,208 meters.

Answer 2

To find the horizontal distance between the point directly beneath the plane and the airport tower, you can use trigonometry. Specifically, the tangent function is useful here because it relates the angle of depression, the opposite side (the altitude of the plane), and the adjacent side (the horizontal distance we're trying to find).

Here's the setup:

The altitude of the plane is 12,000 meters (opposite side of the triangle).
The angle of depression is 32°.
The horizontal distance from the point beneath the plane to the airport tower is
𝑋
X (adjacent side).
We can use the tangent function:

tan
⁡
(
𝜃
)
=
opposite
adjacent
tan(θ)=
adjacent
opposite
​

Substituting the values:

tan
⁡
(
3
2
∘
)
=
12
,
000
𝑋
tan(32
∘
)=
X
12,000
​

Solving for
𝑋
X:

𝑋
=
12
,
000
tan
⁡
(
3
2
∘
)
X=
tan(32
∘
)
12,000
​

I'll calculate this now.

The horizontal distance from the point directly beneath the plane to the airport tower is approximately 19,204 meters. ​

Answer 3

It seems like you're restating the process and calculations for finding the horizontal distance \(X\) from the point directly beneath the plane to the airport tower, using trigonometry. Let's summarize and clarify it once more for accuracy.

Given Data:

• Altitude of the plane: \(12,000\ m\)
• Angle of depression: \(32°\)

Set Up:

Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Here:

• The opposite side is the altitude (12,000 m).
• The adjacent side is the horizontal distance \(X\) from the point directly beneath the plane to the airport tower.

Equation:

Using the angle of depression, we have: \[ \tan(32°) = \frac{12,000}{X} \]

Rearranging to Solve for \(X\):

\[ X = \frac{12,000}{\tan(32°)} \]

Calculation:

Now, we compute \(\tan(32°)\): \[ \tan(32°) \approx 0.6249 \]

Substituting this back into the equation: \[ X = \frac{12,000}{0.6249} \approx 19,207.7 \text{ m} \]

Rounding:

Rounding to the nearest whole number gives: \[ X \approx 19208 \text{ meters} \]

Conclusion:

Thus, the horizontal distance from the point directly beneath the plane to the airport tower is approximately 19,208 meters.

If you have any further questions or need additional clarifications, feel free to ask!