An airplane rises vertically 1000 feet over a horizontal distance of 1 mile. What is the angle of elevation of the airplane's path? (hint: 1 mile = 5280 feet)

Draw a rt triangle and label the hor side 5280ft. Label the ver side 1000ft.

tanA = Y/X = 1000 / 5280 = 0.1894,
A = 10.7 Deg.

To find the angle of elevation, we can use the tangent function.

The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the vertical distance the airplane rises (1000 feet) and the adjacent side is the horizontal distance it covers (1 mile = 5280 feet).

So, let's calculate the tangent of the angle of elevation:

Tangent (θ) = opposite / adjacent
Tangent (θ) = 1000 / 5280

Now we can find the angle (θ) by taking the arctan of both sides:

θ = arctan ( opposite / adjacent )
θ = arctan ( 1000 / 5280 )

Using a calculator, the angle of elevation is approximately 10.6 degrees.

To find the angle of elevation of the airplane's path, we can use the trigonometric definition of the angle of elevation.

The angle of elevation is defined as the angle between the horizontal line and the line of sight when looking upwards. In this case, the line of sight is the path of the airplane.

First, we need to find the total distance traveled by the airplane along its path. Since it rises vertically 1000 feet and travels horizontally 1 mile, we can use the Pythagorean theorem to find the total distance:

Total distance = √(vertical distance² + horizontal distance²)

Plugging in the values, we get:

Total distance = √(1000² + 5280²)

Now, we can find the angle of elevation using the trigonometric function tangent (tan):

tan(angle) = vertical distance / horizontal distance

In this case, the vertical distance is 1000 feet and the horizontal distance is 1 mile (5280 feet). Therefore, we have:

tan(angle) = 1000 / 5280

Now, we can find the angle by taking the inverse tangent (arctan) of both sides:

angle = arctan(1000 / 5280)

Using a calculator, we can find:

angle ≈ 10.67 degrees

Therefore, the angle of elevation of the airplane's path is approximately 10.67 degrees.

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