Problem 58 - Elementary Algebra

Question

A baseball, tossed up into the air from the ground at 32 feet/second, is given by \(h(t)=-16 t 2+32 t\). How long does it take to come back down to the ground?

Step-by-Step Solution

Answer

The baseball returns to the ground after 2 seconds.

Step 1: Identify the Problem

We need to determine the time when the baseball returns to the ground, which means when its height, \(h(t)\), is equal to 0.

Step 2: Set the Equation to Zero

Set the given height equation to zero: \(-16t^2 + 32t = 0\).

Step 3: Factor the Equation

Factor the quadratic equation \(-16t^2 + 32t = 0\). This can be factored as:\[ t(-16t + 32) = 0 \].

Step 4: Solve for t

Using the zero-product property, set each factor equal to zero:\[ t = 0 \] and \[ -16t + 32 = 0 \].

Step 5: Solve the Second Equation

Solve \(-16t + 32 = 0\):Add 16t to both sides:\[ 16t = 32 \]Divide both sides by 16:\[ t = 2 \].

Step 6: Interpret the Solution

The values of \(t\) are 0 and 2. \(t = 0\) represents the initial time when the ball is thrown, and \(t = 2\) is the time it takes for the baseball to return to the ground.

Key Concepts

Factoring

Factoring is a crucial skill when solving quadratic equations. It involves rewriting a quadratic expression as a product of simpler expressions. In the context of the problem, we have the quadratic equation \[-16t^2 + 32t = 0\]. Recognizing that both terms in this equation involve the variable \(t\), we can factor \(t\) out of the equation, resulting in:

• \(t(-16t + 32) = 0\)

This transformation makes it easier to apply the zero-product property. Factoring helps to break down complex problems into simpler pieces, making them much easier to solve.

Zero-Product Property

The zero-product property is a fundamental principle used when dealing with factored quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. After factoring our quadratic equation into:

• \(t(-16t + 32) = 0\)

We set each factor equal to zero:

• \(t = 0\)
• \(-16t + 32 = 0\)

Solving these simple equations gives us the solutions to our problem. This method, leveraging the zero-product property, is efficient and effective for finding solutions quickly and systematically.

Initial Velocity

Initial velocity refers to the speed at which an object begins its journey. In the context of our exercise, it's the speed at which the baseball is thrown upward. Here, the initial velocity is given as 32 feet per second. This value influences the height equation:

• \(h(t) = -16t^2 + 32t\)

This is because the higher the initial velocity, the higher the object goes before coming back down to ground level. Understanding initial velocity is essential for predicting how an object moves, especially in projectile motion scenarios.

Height Equation

A height equation describes the vertical motion of an object under the influence of gravity. In this exercise, the height equation is:

• \(h(t) = -16t^2 + 32t\)

The term \(-16t^2\) accounts for the acceleration due to gravity, which causes the baseball to decelerate as it moves upward and accelerate as it falls back down. The term \(32t\) is influenced by the initial velocity. The height equation allows us to determine the elevation of the object at any point in time and can predict when the object returns to its starting point by setting it equal to zero, as shown in this exercise. Understanding height equations is vital for analyzing the motion of objects under gravity.