Problem 66 - Algebra and Trigonometry

Question

Airplane Trajectory An airplane is flying at a speed of 350 \(\mathrm{milh}\) at an altitude of one mile. The plane passes directly above a radar station at time \(t=0\) . (a) Express the distance \(s\) (in miles) between the plane and the radar station as a function of the horizontal distance \(d\) (in miles) that the plane has flown. (b) Express \(d\) as a function of the time \(t\) (in hours) that the plane has flown. (c) Use composition to express \(s\) as a function of \(t\)

Step-by-Step Solution

Answer

(a) \( s = \sqrt{d^2 + 1} \), (b) \( d = 350t \), (c) \( s(t) = \sqrt{122500t^2 + 1} \).

Step 1: Understand the Problem

The problem involves a right triangle where the airplane is flying horizontally over a radar station. The hypotenuse is the distance from the radar station to the airplane, the height of one mile is one leg of the triangle, and the horizontal distance flown by the airplane is the other leg.

Step 2: Express Distance as a Function of Horizontal Distance

The distance from the radar station to the airplane is the hypotenuse of a right triangle. By Pythagorean theorem, the distance \( s \) is related to the horizontal distance \( d \) and the altitude of 1 mile by \( s = \sqrt{d^2 + 1^2} = \sqrt{d^2 + 1} \).

Step 3: Express Horizontal Distance as a Function of Time

The horizontal distance \( d \) is the product of the speed of the airplane and the time flown: \( d = 350t \), where \( t \) is in hours, and speed is 350 miles per hour.

Step 4: Compose to Express Distance as a Function of Time

Substitute \( d = 350t \) into the expression for \( s \), resulting in \( s(t) = \sqrt{(350t)^2 + 1} = \sqrt{122500t^2 + 1} \).

Step 5: Write Down Final Expression

We now have the function that gives the distance \( s \) between the plane and the radar station in terms of time: \( s(t) = \sqrt{122500t^2 + 1} \).

Key Concepts

Pythagorean Theorem

One of the fundamental mathematical concepts used to solve the airplane trajectory problem is the Pythagorean Theorem. This theorem applies to right triangles and relates the lengths of the sides. Specifically, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula is:\[ c^2 = a^2 + b^2 \] where:

• \(c\) is the length of the hypotenuse
• \(a\) and \(b\) are lengths of the other two sides

In the airplane trajectory problem, the plane's distance from the radar station is the hypotenuse, while the horizontal distance traveled and the fixed vertical altitude act as the two shorter sides. By applying the theorem, you can calculate the direct distance \(s\) from the radar station to the airplane using:\[ s = \sqrt{d^2 + 1} \]This relies on knowing the horizontal distance \(d\) the plane has traveled, and 1 mile as the fixed altitude.

Function Composition

Function composition is a mathematical process where one function is applied to the result of another function. Essentially, you take the output from one function and use it as the input for another, allowing you to express complex relationships in a straightforward manner.In the context of the airplane problem:

• We first found a way to express the horizontal distance \(d\) in terms of time \(t\), which is \(d = 350t\).
• Then, we used the Pythagorean formula \(s = \sqrt{d^2 + 1}\) to express \(s\) in terms of \(d\).

By substituting the expression for \(d\) (which depends on time) directly into the equation for \(s\), function composition lets us establish a relationship between \(s\) and \(t\), resulting in the equation:\[ s(t) = \sqrt{122500t^2 + 1} \]This means that the direct distance from the plane to the radar depends on time in a precise way.

Right Triangle

A right triangle is a triangle with one 90-degree angle. Its properties are simple yet powerful in solving various practical problems, such as the airplane trajectory problem. Understanding the layout of the right triangle helps clarify how distances are measured in such scenarios.In our case:

• The airplane flying over the radar station forms a right triangle.
• The fixed vertical distance from the radar to the plane (altitude) is one leg, which is 1 mile.
• The horizontal distance \(d\), which the plane flies as time progresses, acts as the other leg.
• The hypotenuse, often the most interesting side, represents \(s\), the straight-line distance between the radar station and the airplane.

Using these properties, the problem effectively demonstrates how knowing just initial fixed values and a moving variable allows application of the Pythagorean theorem to determine vital geometrical relationships. This understanding of right triangles plays a key role in determining how distances change as the airplane moves along its flight path.