Question
A plane is descending at a 12 degree angle of depression. If the current altitude of the plane is 1,000 feet, find the distance the plane still needs to fly to reach the ground. Round the answer to the nearest foot. Use the sketchpad to draw a picture.
(Round to the nearest WHOLE NUMBER)
(Round to the nearest WHOLE NUMBER)
Answers
GPT-4o mini
To find the distance the plane still needs to fly to reach the ground while descending at a 12-degree angle of depression from an altitude of 1,000 feet, we can visualize this situation using a right triangle.
1. **Identify the triangle parameters**:
- The altitude (height) of the plane is the vertical side of the triangle (1,000 feet).
- The angle of depression is 12 degrees. From the perspective of the right triangle, this is also the angle between the horizontal distance to the ground and the line of descent.
2. **Using trigonometry**:
In a right triangle, we can use the tangent function which relates the opposite side (the height) and the adjacent side (the horizontal distance).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Rearranging gives us:
\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}
\]
In our case:
- Opposite is the altitude (1,000 feet).
- \(\theta\) is the angle of depression (12 degrees).
3. **Calculate the horizontal distance using the tangent function**:
\[
\text{distance} = \frac{1000}{\tan(12^\circ)}
\]
4. **Using a calculator**:
- First, find \(\tan(12^\circ)\):
\[
\tan(12^\circ) \approx 0.2126
\]
- Now plug this value into the equation:
\[
\text{distance} = \frac{1000}{0.2126} \approx 4691.32 \text{ feet}
\]
5. **Rounding to the nearest whole number**:
The distance the plane needs to fly is approximately **4691 feet**.
In summary, the plane still needs to fly about **4691 feet** to reach the ground.
1. **Identify the triangle parameters**:
- The altitude (height) of the plane is the vertical side of the triangle (1,000 feet).
- The angle of depression is 12 degrees. From the perspective of the right triangle, this is also the angle between the horizontal distance to the ground and the line of descent.
2. **Using trigonometry**:
In a right triangle, we can use the tangent function which relates the opposite side (the height) and the adjacent side (the horizontal distance).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Rearranging gives us:
\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}
\]
In our case:
- Opposite is the altitude (1,000 feet).
- \(\theta\) is the angle of depression (12 degrees).
3. **Calculate the horizontal distance using the tangent function**:
\[
\text{distance} = \frac{1000}{\tan(12^\circ)}
\]
4. **Using a calculator**:
- First, find \(\tan(12^\circ)\):
\[
\tan(12^\circ) \approx 0.2126
\]
- Now plug this value into the equation:
\[
\text{distance} = \frac{1000}{0.2126} \approx 4691.32 \text{ feet}
\]
5. **Rounding to the nearest whole number**:
The distance the plane needs to fly is approximately **4691 feet**.
In summary, the plane still needs to fly about **4691 feet** to reach the ground.