Questions: A small plane is 4 miles by land away from the airport. The angle of elevation from the airport to the airplane is 6.5 degrees. What is the altitude of the plane? Round your answer to the nearest tenth of a mile.

Solution Steps

To find the altitude of the plane, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation to the opposite side (altitude) and the adjacent side (distance by land). The formula is:

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

Rearranging to solve for the opposite side (altitude):

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

Given:

• Distance by land (adjacent) = 4 miles
• Angle of elevation (\(\theta\)) = 6.5 degrees

We can now calculate the altitude.

We are given the following values:

• Distance by land (\(d\)) = 4 miles
• Angle of elevation (\(\theta\)) = \(6.5^{\circ}\)

To use the tangent function, we first convert the angle from degrees to radians: \[ \theta_{\text{radians}} = \frac{6.5 \times \pi}{180} \approx 0.1134 \]

Using the tangent function, we can find the altitude (\(h\)) of the plane: \[ \tan(\theta) = \frac{h}{d} \] Rearranging gives: \[ h = d \cdot \tan(\theta) \] Substituting the known values: \[ h = 4 \cdot \tan(0.1134) \approx 0.4557 \]

Rounding the altitude to the nearest tenth: \[ h \approx 0.5 \text{ miles} \]

Final Answer