An airplane flying at an altitude of 7200 feet sights the angle of depression to a control tower to be 18°.
Find the horizontal distance the plane is from the control tower.
(Disregard the height of the tower and round your answer to the nearest foot.)
To find the horizontal distance, we need to apply trigonometry.
Let's denote the horizontal distance from the plane to the control tower as x.
We know that the angle of depression is 18°, which means that the angle between the horizontal line and the line of sight from the plane to the control tower is also 18°.
Now we can use the tangent function (tan) to find x:
tan(18°) = opposite/adjacent
In this case, the opposite side is the altitude of the plane, which is 7200 feet. The adjacent side is the horizontal distance x, which we want to find.
Using the formula:
tan(18°) = 7200/x
To solve for x, we can rearrange the equation:
x = 7200 / tan(18°)
Using a calculator:
x ≈ 24999.2 feet
Rounding to the nearest foot, the horizontal distance from the plane to the control tower is 25000 feet.
Possible answers:
A)
≈ 2,339 ft
B)
≈ 23,301 ft
C)
≈ 22,159 ft
D)
≈ 7,570 ft
None of the options provided matches the calculated value.
Using the trigonometric equation:
x = 7200 / tan(18°)
x ≈ 24999.2 feet
Therefore, rounding to the nearest foot, the correct answer should be:
E) ≈ 25000 ft
To find the horizontal distance from the control tower to the plane, we can use trigonometry. Given the angle of depression and the altitude of the plane, we can use the tangent function.
1. Let's start by labeling the relevant information in the problem:
- Angle of depression = 18°
- Altitude of the plane = 7200 feet
2. We can draw a right triangle to represent the situation. The altitude of the plane is the opposite side, and the horizontal distance is the adjacent side.
3. We can use the tangent function to find the horizontal distance:
tan(angle of depression) = opposite/adjacent
Plugging in the given values:
tan(18°) = 7200/adjacent
4. To find the adjacent side, we need to isolate it. Rearranging the equation:
adjacent = 7200 / tan(18°)
5. Using a calculator, find the tangent of 18°:
tan(18°) ≈ 0.3249
6. Substitute this value back into the equation to find the horizontal distance:
adjacent ≈ 7200 / 0.3249
7. Calculating this value:
adjacent ≈ 22180.8847
8. The horizontal distance the plane is from the control tower is approximately 22181 feet.
Therefore, the plane is approximately 22181 feet away from the control tower.