Problem 43 - College Algebra

Question

Fireworks are launched into the air. The quadratic function $$s(t)=-16 t^{2}+200 t+4$$ models the fireworks' height, \(s(t),\) in feet, \(t\) seconds after they are launched. When should the fireworks explode so that they go off at the greatest height? What is that height?

Step-by-Step Solution

Answer

The fireworks should explode at 6.25 seconds after they are launched to achieve the maximum height of 631.25 feet.

Step 1: Identify the quadratic function

The quadratic function provided is \( s(t) = -16t^{2} + 200t + 4 \). We can easily see that it is in the form \( ax^{2} + bx+ c \) whereby a, b, and c represent coefficients.

Step 2: Find the time when the height is maximum

The maximum value of a quadratic function is found at its vertex. The x-coordinate (in this case \( t \) ) of the vertex of a quadratic function is given by the formula \( -b/(2a) \). Using the coefficients from the function \( s(t) \), we get \( t = -200/(-2 \times16) = 6.25 \) seconds

Step 3: Find the maximum height

The maximum height is the value \( s(t) \) at the time \( t \) found in step 2. Substituting \( t = 6.25 \) into \( s(t) \) gives \( s(6.25) = -16(6.25)^{2} + 200 \times 6.25 + 4 \), calculating this we get \( 631.25 \) feet.

Key Concepts

Vertex of a Quadratic Function

Understanding the vertex of a quadratic function is crucial to solve various types of problems, including those involving projectile motion like the path of fireworks. A quadratic function is generally represented in the standard form of: \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this representation, the vertex is the highest or lowest point on the graph, known as the graph's peak or trough.

The coordinates of the vertex \( (h, k) \) can be calculated using the formula \( h = -\frac{b}{2a} \) and then evaluating \( k \) by substituting \( h \) back into the function. The vertex represents the maximum or minimum value of the quadratic function depending on the sign of the coefficient \( a \). If \( a \) is positive, the parabola (the graph's shape) opens upwards, and the vertex is the minimum point. If \( a \) is negative, as in the fireworks problem, the parabola opens downwards, and the vertex signifies the maximum height the fireworks can reach.

Knowing how to find the vertex is not only advantageous for academic exercises but is also practical in real-life scenarios where prediction and optimization are necessary.

Maximum Value of a Quadratic

When working with quadratic functions such as \( y = ax^2 + bx + c \), where \( a < 0 \), the maximum value is a point of great interest, particularly when dealing with questions of optimization. The maximum value of a quadratic function is found at its vertex, as previously explained.

The maximum value is significant in many real-world situations, like determining the best price to sell an item to maximize profit or, as in the context of our exercise, finding the optimal time to detonate fireworks to achieve the greatest spectacle. The maximum value is effectively the \( y \)-coordinate of the vertex (\( k \)), which you determine after finding the \( x \)-coordinate (\( h \)) from the vertex's formula. Once \( h \) is found, you substitute it into the original equation to get \( k \), which is the maximum value of the quadratic function.

It's noteworthy that the ability to calculate the maximum value has important applications in various fields such as economics, engineering, and even sports analytics, where determining the peak performance or value is essential.

Quadratic Function Applications

Quadratic functions are not just theoretical constructs but have myriad applications in the real world. From physics to finance, these functions help us model and solve practical problems. In the given exercise, a quadratic function is employed to model the trajectory of fireworks, which is a quintessential example of the application of quadratic functions in physics.

Projectile motion, object's paths, the area of land plots, and even market trends in finance can be analyzed using quadratic functions. More examples include:

• Physics: Determining the trajectory of a ball thrown into the air or the optimal angle for a satellite dish.
• Engineering: Designing parabolic arches and suspension cables in bridges.
• Biology: Modeling the spread of diseases or population growth under specific conditions.
• Economics: Analyzing the relationship between supply-demand and pricing.

These real-life applications underline the importance of mastering quadratic functions for students who will apply these principles in their future careers. The functions provide a powerful tool for predicting outcomes and making informed decisions in a variety of disciplines.