Problem 67 - Introductory Algebra for College Students

Question

A ball is thrown straight up from a rooftop 300 feet high. The formula $$h=-16 t^{2}+20 t+300$$ describes the ball's height above the ground, \(h,\) in feet, \(t\) seconds after it was thrown. The ball misses the rooftop on its way down and eventually strikes the ground. The graph of the formula is shown, with tick marks omitted along the horizontal axis. Use the formula to solve Exercises \(67-69\) (GRAPH NOT COPY) How long will it take for the ball to hit the ground? Use this information to provide tick marks with appropriate numbers along the horizontal axis in the figure shown.

Step-by-Step Solution

Answer

It will take approximately 5.63 seconds for the ball to hit the ground.

Step 1: Set the Formula Equal to 0

Given the formula \(h=-16 t^{2}+20 t+300\), set \(h=0\) to find out the time the ball hits the ground. This gives: \[0=-16 t^{2}+20 t+300\]

Step 2: Solve the Quadratic Equation

The equation can be solved by applying the quadratic formula \(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In the equation, the coefficients are \(a= -16\), \(b=20\), and \(c=300\). Solving will give two roots for \(t\).

Step 3: Choose the Appropriate Root

Since time cannot be negative, choose the root that is a positive number.

Step 4: Plot on the Graph

Finally, use the time value as the tick marks for the horizontal axis on the graph.

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone of algebra and a fundamental tool in solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). Trusted by many, the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) not only delivers the solutions for \(x\) but also demystifies how roots relate to the coefficients.

In the context of the given exercise, the formula unveils the times when the ball will reach ground level, which corresponds to the height (\(h\)) being zero. By identifying \(a = -16\), \(b = 20\), and \(c = 300\), and substituting these into the quadratic formula, one can calculate the roots. The positive root, which represents a meaningful duration in time units, will answer the question of when the ball strikes the ground. Understanding the quadratic formula is pivotal for students not only to solve the problem but also to comprehend the relationship between the physical situation and its mathematical representation.

Projectile Motion

Projectile motion is a form of motion experienced by an object that is launched into the air and moves under the influence of gravity alone. The path of a projectile is a parabola, and its motion can be predicted using kinematic equations. In our problem, a ball thrown from a rooftop follows this kind of trajectory.

Formulas for projectile motion often take the form of a quadratic equation, like \(h = -16t^2 + 20t + 300\), reflecting the gravity's impact (\(16t^2\)) and initial velocity (\(20t\)). Recognizing that \(h = 0\) when the ball hits the ground enables us to solve for the time (\(t\)) using the aforementioned quadratic formula. Through this lens, students can see physics concepts exemplified within a real-world scenario and grasp how quadratic equations model the behavior of objects in motion, capturing the essence of projectile dynamics.

Graphing Quadratic Functions

Graphing quadratic functions is a visual approach to understanding the behavior of quadratic equations. A typical quadratic function graphs into a curve called a parabola, which can open either up or down depending on the sign of the leading coefficient.

When graphing the function \(h = -16t^2 + 20t + 300\), one would expect a downward-opening parabola because the coefficient of \(t^2\) is negative. The graph provides valuable insights into the trajectory the ball takes after being thrown. Important features to plot include the vertex, representing the peak height of the ball, and the roots—or the x-intercepts—indicating when the ball will reach the ground level. For students addressing the given exercise, accurately placing tick marks on the horizontal axis to reflect the roots is essential. These marks are derived from the times calculated using the quadratic formula. A clear graph aids in visualizing when precisely the ball will make contact with the ground, enriching a student's comprehension of quadratic functions and their graphs in relation to physical phenomena.