Question

As an airplane descends toward an airport, it drops a vertical distance of \(24 \mathrm{~m}\) and moves forward a horizontal distance of \(320 \mathrm{~m}\). What is the distance covered by the plane during this time?

Short answer

The plane covers approximately 320.86 m.

Step-by-step solution

Understanding the Right Triangle

The airplane's movement creates a right triangle where the vertical drop of 24 m is one leg and the horizontal forward movement of 320 m is the other leg. The distance covered by the airplane is the hypotenuse of this right triangle.

Using the Pythagorean Theorem

We apply the Pythagorean Theorem to find the hypotenuse. The formula is given by: \[ c = \sqrt{a^2 + b^2} \] where \( a \) is the vertical drop and \( b \) is the horizontal movement.

Substituting the Known Values

Let's substitute the known values into the formula: \[ c = \sqrt{(24)^2 + (320)^2} \] This simplifies to: \[ c = \sqrt{576 + 102400} \]

Calculating the Hypotenuse

Continue the calculation: \[ c = \sqrt{102976} \] Using a calculator, find \( c \approx 320.86 \).

Interpreting the Result

The airplane covers a distance of approximately 320.86 meters as it descends.

Key concepts

Right Triangle

A right triangle is a special type of triangle that has one angle precisely 90 degrees. In these triangles, the side opposite the right angle is the longest and is called the hypotenuse. The other two sides are known as the legs. When an object moves along two perpendicular paths—one vertical and one horizontal—a right triangle is formed. Think of walking down steps diagonally where your descent forms one leg, your forward step forms another, and your path forms the hypotenuse.
This is exactly what happens when an airplane descends. As it moves horizontally forward and simultaneously drops vertically, these movements represent the legs of a right triangle. Thus, understanding right triangles allows us to assess the airplane's total descent path accurately.

Distance Calculation

Calculating the distance in a right triangle involves using the Pythagorean Theorem. This theorem is a mathematical equation stating that in a right triangle, the square of the hypotenuse length is equal to the sum of the squares of the lengths of the other two sides.

• Formula: \( c^2 = a^2 + b^2 \)
• Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) are the triangle's legs.

For instance, with our airplane scenario, where \( a = 24 \, \text{m} \) and \( b = 320 \, \text{m} \), we substitute these values to find \( c \) as follows:
\[ c = \sqrt{(24)^2 + (320)^2} \].
This simplifies to \( c = \sqrt{102976} \), giving the approximate distance the airplane covers. Using a calculator, the plane's descent path comes out to approximately 320.86 meters.

Vertical and Horizontal Components

The vertical and horizontal components are crucial in many distance calculations. They represent how far something moves in a straight line up or down, and how far it goes straight ahead.
For the plane example, the vertical component is the drop or descent of 24 meters, and the horizontal component is the forward movement of 320 meters. These components together help us form a complete picture of movement, as they highlight two key directions of any object's path.
Understanding these allows us to effectively use the Pythagorean Theorem to piece together the complete distance traveled. Recognizing the components' roles ensures not just accurate calculations but also enhances spatial awareness of movement in real-world physics problems.