Problem 35 - Elementary and Intermediate Algebra

Question

Quadratic Equation Model Problems Thrill Rides. At the peak of a roller coaster ride, a rider's sunglasses fly off his head. The height \(h\) (in feet) of the glasses, \(t\) seconds after he loses them, is given by \(h=-16 t^{2}+64 t+80 .\) After how many seconds will the glasses hit the ground? (Hint: Factor out - \(16 .\) )

Step-by-Step Solution

Answer

The glasses will hit the ground after 5 seconds.

Step 1: Understand the Problem

You are given the height equation of the sunglasses as a function of time: \( h = -16t^2 + 64t + 80 \). The glasses hit the ground when the height \( h \) is 0. We need to find the value of \( t \) for which \( h = 0 \).

Step 2: Set the Equation to Zero

Set \( h = 0 \) in the equation to find when the glasses reach the ground. The equation becomes \( 0 = -16t^2 + 64t + 80 \).

Step 3: Factor Out Common Factor

Look for a common factor in the equation. Notice that each term is divisible by \(-16\). Factor it out: \(-16(t^2 - 4t - 5) = 0 \).

Step 4: Solve the Quadratic Equation

The equation simplifies to \( t^2 - 4t - 5 = 0 \). We'll factor the quadratic to solve for \( t \). We need two numbers that multiply to \(-5\) and add to \(-4\). These numbers are \(-5\) and \(1\). Thus, \( (t - 5)(t + 1) = 0 \).

Step 5: Find the Solutions for t

Using the factored form \((t - 5)(t + 1) = 0\), set each factor equal to zero: \(t - 5 = 0\) or \(t + 1 = 0\). Solve these to find \( t = 5 \) or \( t = -1 \).

Step 6: Select the Relevant Solution

The time \( t \) must be non-negative because it represents seconds after the glasses are released. So, \( t = -1 \) is not a valid solution. Therefore, \( t = 5 \) seconds is the correct answer.

Key Concepts

Projectile Motion

In problems involving projectile motion, like our roller coaster sunglasses problem, we often see objects flying through the air under the influence of gravity. When we say projectile motion, we mean motion in a parabolic path. This pattern is due to the influence of two main forces: the force exerted by the object itself and the gravitational force pulling it down.

Key points about projectile motion include:

• The motion is characterized by a parabolic trajectory.
• Gravity constantly acts on the object, pulling it downward.
• In an ideal situation, where air resistance is negligible, the only force acting on the object is gravity.

In our sunglasses model, the equation describes how high the sunglasses are at a given time after being released. Over time, as gravity works its magic, the height decreases until it hits zero (the ground). This entire operation shows the elegant, predictable nature of projectile motion.

Factoring Quadratics

Factoring quadratics is a fundamental skill in algebra that we use when simplifying polynomial expressions. When you factor a quadratic, you're finding two binomials that multiply together to yield the original quadratic expression.

Let’s explore this further in steps:

• Identify the quadratic equation. In our exercise: \( -16(t^2 - 4t - 5) = 0 \).
• Factor out any common factors if possible. Here, that common factor was \(-16\).
• Now, focus on factoring the quadratic inside the parentheses: \(t^2 - 4t - 5\).
• Find two numbers that multiply to give the constant term (\(-5\)) and add to give the middle coefficient (\(-4\)). These numbers are \(-5\) and \(1\).
• The quadratic factors to \((t - 5)(t + 1) = 0\).

Learning to factor quadratics is akin to breaking a complicated situation into smaller, more manageable steps. This skill is extremely useful across many areas in mathematics.

Word Problems

Word problems offer practical contexts in math problems, allowing us to apply mathematical equations and logic to real-world scenarios. Despite their intimidations, they teach important skills such as critical thinking and problem-solving.

To tackle word problems, use this strategy:

• Read the problem carefully and identify what is being asked. Understand the scenario before jumping into calculations.
• Translate words into a mathematical equation. Recognize the given data and relate them to mathematical expressions, as with our height equation \(h = -16t^2 + 64t + 80\) for the sunglasses.
• Set the equation to resolve the problem. This might involve setting things to zero or expressing one quantity in terms of another, like finding when the height \(h\) becomes zero.

Practice helps in efficiently converting real-life problems into mathematical language, ultimately offering a deeper insight into how mathematics models reality.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. They’re significant in mathematics for their versatility and the intuitive way they model a wide variety of situations.

Consider these aspects of polynomial functions:

• A polynomial function encompasses terms of various degrees. For our equation \(h = -16t^2 + 64t + 80\), the degree is 2 because the highest power is \(t^2\).
• The coefficients in these functions show the strength or effect of each term. Here, \(-16\), \(64\), and \(80\) are coefficients representing initial velocity, linear time influence, and initial height in our problem.
• The graph of a quadratic polynomial is a parabola. Depending on the sign of the leading coefficient (\(-16\) in this case), the parabola opens up or down.

Understanding polynomial functions enables us to predict their behavior, find zeros, and apply these insights to solve both theoretical and practical problems.