Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Two planes leave simultaneously from the same airport, one flying due east and the other due south. The eastbound plane is flying 100 miles per hour faster than the southbound plane. After 2 hours the planes are 1500 miles apart. Find the speed of each plane.

Step-by-Step Solution

Answer

The speed of the southbound plane is approximately 570 miles per hour, and the speed of the eastbound plane is approximately 670 miles per hour.

Step 1 - Determine the speeds

Denote the speed of the southbound plane as \(x\) miles per hour. According to the problem, the eastbound plane is flying 100 miles per hour faster than the southbound plane, so the speed of the eastbound plane is \(x+100\) miles per hour.

Step 2 - Formulate the equation

Each plane has been flying for 2 hours, so the distance each plane has traveled is its speed multiplied by the time. This gives us \(2x\) miles for the southbound plane and \(2(x+100)\) miles for the eastbound plane. Since these distances form a right triangle with the distance between the planes, we can use the Pythagorean theorem to write an equation: \( (2x)^2 + (2(x+100))^2 = 1500^2 \).

Step 3 - Simplify and solve the equation

Expand and simplify the equation to get: \(4x^2 + 4(x^2 + 200x + 10000) = 2250000\). Further simplify it to: \(8x^2 + 800x + 40000 = 2250000\). Rearrange this into a standard quadratic equation: \(8x^2 + 800x - 2210000 = 0\). Divide the equation by 8 to make it easier to solve: \(x^2 + 100x - 276250 = 0\). Solve this quadratic equation using the quadratic formula: \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). Substituting the coefficients gives the solution for \(x\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, commonly used when solving problems that involve right triangles. In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed mathematically as:

• For a right triangle with sides of lengths \(a\) and \(b\), and hypotenuse \(c\), the theorem states that \(a^2 + b^2 = c^2\).

The exercise involving the two planes uses the Pythagorean Theorem because the paths the planes take can be thought of as two sides of a right triangle, with the distance between them being the hypotenuse. In this context:

• The eastbound plane travels in one direction, and the southbound plane travels perpendicular to that direction.
• Hence, their traveled paths can form a right triangle where the hypotenuse is the distance between the planes after 2 hours, which is given as 1500 miles.

Utilizing the Pythagorean Theorem, you can form an equation relating the distances traveled by the planes and solve for the speed of each plane.

Distance, Rate, and Time Problems

Distance, rate, and time problems are common in algebra and physics as they help to model real-world scenarios involving movement. The basic relationship among these quantities is given by the formula: \( \text{Distance} = \text{Rate} \times \text{Time}\). This formula states that the distance traveled by an object is equal to its speed (or rate) multiplied by the time spent traveling.

• For instance, if a car travels at a speed of 60 miles per hour for 3 hours, the distance traveled is \(60 \times 3 = 180 \) miles.

In the problem with the two planes, each plane's speed and time of travel enable you to find out how far they traveled individually.

• Let \(x\) be the speed of the southbound plane in miles per hour.
• Thus, in 2 hours, it travels \(2x\) miles.
• The eastbound plane travels faster by 100 miles, making its speed \(x + 100\) miles per hour, traveling \(2(x + 100)\) miles in the same duration.

Establishing these distances allows you to use them as legs of the right triangle when applying the Pythagorean Theorem.

Problem Solving in Algebra

Algebra offers powerful tools for solving a wide variety of problems, often involving the setup of equations based on given scenarios. In these scenarios, you identify unknowns, express the given information as mathematical equations, and utilize algebraic methods to find solutions.In this problem:

• Identify the variables: Recognize what you need to find. Here, it's the speeds of the two planes.
• Translate the problem into an equation: Use relationships such as distance, rate, and time along with geometric principles like the Pythagorean Theorem.
• Create the quadratic equation: From the scenario, derive a quadratic equation relating to these speeds.
• Solve the equation: Typically, either factoring, completing the square, or using the quadratic formula.

The given problem boils down to solving a quadratic equation like \(x^2 + 100x - 276250 = 0\). Here, you apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the coefficients \(a = 1\), \(b = 100\), and \(c = -276250\) into this formula yields the speeds of the southbound and eastbound planes.