Question

Flight Control An airplane is flying in still air with an airspeed of 275 miles per hour. The plane is climbing at an angle of \(18^{\circ} .\) Find the rate at which it is gaining altitude.

Short answer

The airplane is gaining altitude at a rate of approximately 86.3 miles per hour.

Step-by-step solution

Evaluate the given details

We know that the airplane is flying with an airspeed of 275 miles per hour at an angle of \(18^{\circ}\). We are required to find the rate at which it is gaining altitude. The airspeed is the hypotenuse of the triangle, the rate of altitude gain is the opposite side and the angle is \(18^{\circ}\).

Use sine function

In a right triangle, sin of an angle can be described as the ratio of the length of the opposite side to the length of the hypotenuse. Hence, we can represent the problem using the sine function: \(sin(18^{\circ}) = opposite / 275\)

Solve for opposite

Rearrange the equation from Step 2 to solve for 'opposite': \(opposite = 275 * sin(18^{\circ})\)

Evaluate the above expression

Use a calculator to compute the value of sin(18), and then multiply by 275.

Interpret the result

The numerical value obtained in Step 4 represents the rate of altitude gain in miles per hour.

Key concepts

Airspeed Calculation

Understanding airspeed is crucial in aviation, as it is the speed at which an aircraft moves through the still air. It is important to note that airspeed differs from ground speed, which is affected by wind conditions. In our context, the airspeed of 275 miles per hour serves as the hypotenuse of a right triangle in a trigonometric problem. The speed through the air helps determine climbing patterns and angles - key aspects of safe and efficient flight. For an airplane climbing at an angle, the airspeed aligns with the hypotenuse of the triangle formed by the climb. Knowing this, we can find other components like the rate of altitude gain through trigonometric relationships, which we will explore further in the following sections.

Sine Function

The sine function is a foundational concept in trigonometry, often encountered as "SOH" in the SOH-CAH-TOA mnemonic. It denotes the ratio of the length of the side opposite a given angle to the hypotenuse of a right-angled triangle. The sine function's formula is expressed as: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). This is precisely how we use the sine function in airspeed and altitude calculations. For instance, when the airplane climbs at an angle of \(18^{\circ}\), the sine function enables us to express the relationship between the airspeed of 275 miles per hour and the rate of altitude gain (opposite side). The connection via sine helps us translate angular motion into vertical movement, a critical step in determining altitude changes.

Altitude Rate Calculation

Calculating the rate of altitude gain requires translating the motion at an angle to vertical speed. This is where the sine function becomes instrumental. By rearranging the sine expression: \( \sin(18^{\circ}) = \frac{\text{opposite}}{275} \), we solve for the 'opposite' side, which represents the altitude gain. By isolating 'opposite', we arrive at: \( \text{opposite} = 275 \times \sin(18^{\circ}) \). This simplified equation can be computed using a calculator to obtain a numerical value, indicating the altitude rate in miles per hour. This calculation is vital for pilots to plan climbs effectively, ensuring both optimal speed and safety.