Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Hitting a Baseball A baseball is hit so that its height in feet after \(t\) seconds is given by $$s(t)=-16 t^{2}+44 t+4$$ (a) How high is the baseball after 1 second? (b) Find the maximum height of the baseball.

Step-by-Step Solution

Answer

(a) The baseball is 32 feet high after 1 second. (b) The maximum height is approximately 53.25 feet.

Step 1: Substitute to Find Height After 1 Second

To find out how high the baseball is after 1 second, substitute \( t = 1 \) into the height equation \( s(t) = -16t^2 + 44t + 4 \). Compute \( s(1) = -16(1)^2 + 44(1) + 4 \).

Step 2: Simplify the Expression

Calculate the expression from Step 1: \( s(1) = -16(1) + 44 + 4 = -16 + 44 + 4 \). Simplify this to find the height after 1 second.

Step 3: Solve for the Maximum Height - Find the Vertex

The quadratic function \( s(t) = -16t^2 + 44t + 4 \) represents a parabola opening downwards. The maximum height is at the vertex. Use the formula \( t = -\frac{b}{2a} \) where \( a = -16 \) and \( b = 44 \) to find the time \( t \).

Step 4: Calculate Time at Maximum Height

Substitute \( a = -16 \) and \( b = 44 \) into \( t = -\frac{b}{2a} \) resulting in \( t = -\frac{44}{2(-16)} = \frac{44}{32} = \frac{11}{8} \) seconds.

Step 5: Substitute to Find Maximum Height

Now substitute \( t = \frac{11}{8} \) back into the height equation: \( s\left(\frac{11}{8}\right) = -16\left(\frac{11}{8}\right)^2 + 44\left(\frac{11}{8}\right) + 4 \). Simplify the expression to find the maximum height.

Key Concepts

Parabolic Motion

Parabolic motion describes the path an object follows when it is launched or thrown and its trajectory is affected by gravity. A baseball being hit into the air is a perfect example. When the ball is hit, it moves in a curved path due to the forces acting on it, namely the initial velocity and gravity. This path can be described by a quadratic equation like the one given in the exercise:

• The general form of a quadratic equation is: \( s(t) = at^2 + bt + c \), where \( s(t) \) provides the height of the object at a given time \( t \).
• In the exercise, the equation \( s(t) = -16t^2 + 44t + 4 \) shows how the ball's height changes over time as a result of the parabolic motion.
• The negative coefficient of \( t^2 \) indicates the parabola opens downward, which is typical for objects influenced by gravity like a baseball.

Understanding parabolic motion helps us calculate important details such as where and when an object will reach its maximum height, and eventually, when and where it will land. In solving these problems, the concepts of vertex of a parabola and the associated maximum height become crucial.

Vertex of a Parabola

The vertex of a parabola is a key feature for understanding quadratic equations, especially in scenarios involving objects launched into the air. Since the quadratic equation in the exercise represents the path of a baseball, identifying the vertex lets us find the maximum height of the object.

For a parabola represented by \( ax^2 + bx + c \), the vertex can be calculated using the formula:
\[ t = -\frac{b}{2a} \]Here, 't' is the time at which the baseball reaches its maximum height. In this exercise:

• \( a = -16 \) and \( b = 44 \)
• Substituting into the formula gives \( t = -\frac{44}{2(-16)} = \frac{44}{32} = \frac{11}{8} \) seconds.

This indicates the ball reaches its maximum height at \( \frac{11}{8} \) seconds after being hit. Finding the vertex allows us to predict where on the path the highest point will occur, an essential part of analyzing projectile motion.

Maximum Height

Once the time at which the object reaches its maximum height is known, calculating this height involves substituting the time back into the original quadratic equation. It's crucial for understanding how high an object will go during its trajectory.

In the exercise, substituting \( t = \frac{11}{8} \) into the height equation:
\[ s\left(\frac{11}{8}\right) = -16\left(\frac{11}{8}\right)^2 + 44\left(\frac{11}{8}\right) + 4 \]

• First, calculate \( \left(\frac{11}{8}\right)^2 = \frac{121}{64} \)
• Then compute each part of the multiplication and addition operations.
• The result of this computation gives the maximum height where the baseball reaches its peak in the air.

Understanding this concept not only answers the specific problem but also helps learners grasp how quadratic equations model real-world situations, such as the flight of a baseball, by showing its highest point and other aspects of its motion.