Problem 66 - Elementary Algebra

Question

The height of a projectile launched straight up from a mound is given by the function \(h(t)=-16 t 2+96 t+4,\) where \(t\) represents seconds after launch. What is the maximum height?

Step-by-Step Solution

Answer

The maximum height is 148 feet.

Step 1: Identify the Quadratic Function

The function given is a quadratic function in the form \( h(t) = -16t^2 + 96t + 4 \). In this form, \( a = -16 \), \( b = 96 \), and \( c = 4 \).

Step 2: Determine the Vertex Formula

The vertex of a quadratic function \( ax^2 + bx + c \) provides the maximum (or minimum) point. The \( t \)-coordinate of the vertex is given by \( t = -\frac{b}{2a} \).

Step 3: Calculate the Time at Maximum Height

Substitute the known values into the vertex formula: \[ t = -\frac{96}{2(-16)} = \frac{96}{32} = 3 \] This means the projectile reaches its maximum height at \( t = 3 \) seconds.

Step 4: Substitute Time Back into the Height Function

Now, substitute \( t = 3 \) back into the height function to find the maximum height: \[ h(3) = -16(3)^2 + 96(3) + 4 \] \[ h(3) = -16(9) + 288 + 4 \] \[ h(3) = -144 + 288 + 4 \] \[ h(3) = 148 \] So, the maximum height is 148 feet.

Key Concepts

Vertex Formula

In the world of quadratic functions, the vertex formula is a handy tool that helps us pinpoint the highest or lowest point on the graph of a parabola. For a quadratic function written in the form \[ h(t) = at^2 + bt + c \]we determine the vertex using the formula\[ t = -\frac{b}{2a} \]This formula is derived from the process of "completing the square," and it essentially tells us where the tip or "vertex" of the parabola is, in terms of its horizontal position on a graph. Once we know this vertex location, we can easily calculate the corresponding height or depth, depending on the context of the problem. In projectile motion problems, the vertex represents the maximum height the object reaches. It’s vital to understand this concept, as it allows the prediction of peak points in trajectories.

Projectile Motion

Projectile motion refers to the path that an object follows when it is thrown or propelled into the air, affected only by the force of gravity. This kind of motion is common in sports and other everyday settings, like throwing a ball or launching a rocket. In this exercise, the function \[ h(t) = -16t^2 + 96t + 4 \]gives us a mathematical model to predict the height of a projectile at any given time.

• The term \(-16t^2\) represents the effect of gravity pulling the projectile downward.
• The term \(96t\) reflects the initial velocity of the projectile moving upward.
• The term \(4\) is the initial height from which the projectile is launched.

Understanding the basic components of the equation helps in figuring out how different forces are acting upon the object at any given time. With the help of these parameters, we can predict the projectile's trajectory and its varying height over time.

Maximum Height

The concept of maximum height in projectile motion is vital because it indicates the peak of an object's trajectory. Once an object is thrown upwards, it eventually slows down, stops at its peak or maximum height, then descends due to gravity. In our current exercise, the maximum height is achieved at the vertex of the quadratic function. We already calculated that this height occurs at \(t = 3\) seconds. By substituting back into the equation, we determined the height is\[ h(3) = -16(3)^2 + 96(3) + 4 = 148 \]The highest point the projectile reaches is 148 feet. This calculation matters in real-world applications, from athletics to engineering, confirming expectations about how high a projectile will travel, ensuring safety, and even for the enjoyment of seeing how things fly!