Problem 31 - College Algebra Essentials

Question

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{cc} \hline x, \text { seconds after the ball is } & y, \text { ball's height, in feet, above } \\ \text { thrown } & \text { the ground } \\ \hline 1 & 224 \\ 3 & 176 \\ 4 & 104 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

Step-by-Step Solution

Answer

The quadratic function we obtain is \(y = 16x^2 - 32x + 240\). When \(x = 5\), the function gives \(y = 200\), meaning the ball is 200 feet above the ground after 5 seconds.

Step 1: Developing the Equations

The quadratic function \(y = ax^2 + bx + c\) should be satisfied by all the data points. Thus, we generate three equations by inserting the x and y values from the table into the equation. This gives us: \\ Equation 1 from (x=1, y=224): \(224 = a + b + c\) \\ Equation 2 from (x=3, y=176): \(176 = 9a + 3b + c\) \\ Equation 3 from (x=4, y=104): \(104 = 16a + 4b + c\)

Step 2: Solving the simultaneous equations

These three equations can be solved simultaneously. If we subtract equation 1 from equation 2, we get: \(176 - 224 = 9a - a + 3b - b\), which simplifies to \(-48 = 8a + 2b\). Likewise, subtracting equation 1 from equation 3 gives us \(-120 = 15a + 3b\). Solving these two equations produces a=16 and b=-32. Lastly, plugging these values into the first equation yields c=240.

Step 3: Determining the Quadratic Function

Substituting the found values of a, b and c into the quadratic function gives us the function \(y = 16x^2 - 32x + 240\)

Step 4: Calculating y for x=5

Substitute \(x = 5\) into our newly found equation: \(y = 16(5)^2 - 32(5) + 240\), which results in \(y = 200\). This means the ball will be 200 feet above the ground 5 seconds after it was thrown.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with different variables. In the context of our exercise, we formulate a system based on the quadratic function y = ax^2 + bx + c, representing the ball's trajectory. Given data points serve as inputs to establish equations, which are then solved simultaneously to find the unknown coefficients a, b, and c.

This method of solving for multiple unknowns is vital in algebra and helps students understand the interconnected nature of variables. By practicing such problems, learners enhance their ability to think critically and solve complex real-world problems.

Parabolic Motion

Parabolic motion describes the trajectory of an object thrown in the air and affected only by gravity. This motion is parabolic because the graph of height versus time is a parabola. In the given problem, a ball thrown from a rooftop follows such a path, characterized by a quadratic function.

Understanding this concept not only helps in math but also in physics, as it is a fundamental principle behind projectile motion. By analyzing the coefficients a, b, and c in the quadratic equation, students can infer various attributes of the motion like the initial velocity, maximum height, and the time it will take to reach the ground.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. The quadratic function from our exercise, y = ax^2 + bx + c, is a classic example. It models the height y of the ball at any time x during its fall.

Learning to manipulate these expressions is fundamental to algebra and higher mathematics. By substituting known quantities into an algebraic expression, we can solve for unknown variables, which in turn aids in understanding more complex problems. Furthermore, simplifying and factoring algebraic expressions are essential skills for any math student.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world scenarios to make predictions or understand phenomena. The exercise showcases mathematical modeling by using a quadratic function to predict the ball's height at any given time after being thrown.

This process consists of identifying relevant variables, constructing functions to describe their relationships, and then using the model to analyze or simulate real-world behavior. Such skills are increasingly important in various fields, such as engineering, economics, and environmental science, making this an essential concept for students to grasp.