Question

At Buckingham Fountain in Chicago, the height \(h\) (in feet) of the water above the main nozzle can be modeled by \(h=-16 t^2+89.6 t\), where \(t\) is the time (in seconds) since the water has left the nozzle. Describe three different ways you could find the maximum height the water reaches. Then choose a method and find the maximum height of the water.

Short answer

The maximum height is found by first applying the formula for the vertex of a parabola \(t = -b / (2a)\), then substituting this 't' back into the original function to get the maximum 'h' value. Other methods could include graphing the function or completing the square.

Step-by-step solution

Understand the problem

Here we are given a function of height depending on time. This is a quadratic equation with time 't' as the variable and 'h' as the function of time. The maximum height is the maximum value of this function, which is the vertex of the parabolic curve described by the function.

Finding the vertex using the algebraic method

The maximum point or vertex of the parabolic curve can be found using the formula \(t = -b / (2a)\). Here in the equation \( h = -16t^2 + 89.6t \), 'a' is -16 and 'b' is 89.6. Applying the formula, the maximum point is achieved at time \( t = -89.6 / (2 * -16) \).

Calculate the maximum height

The time at which the maximum height is achieved has been calculated in the previous step. Substituting this value of 't' in the equation for 'h', we get the maximum height as \( h_{max} = -16t_{max}^2 + 89.6t_{max} \).

Alternative methods

Other ways to find the maximum height could include graphing the function or applying the completion of square method. For the graphing method, simply plot the function and find the maximum point. For the 'completing the square' method, you would rearrange the equation into the form \(h=a(t-h)^2+k\), where \( (h,k) \) would be the vertex of the parabola.

Key concepts

Parabolic Curves

A parabolic curve is the graphical representation of a quadratic function, which looks like a U-shaped or an inverted U-shaped symmetrical curve. This curve is known as a "parabola." Parabolas differ based on the coefficient of the squared term in the quadratic equation. If the coefficient is positive, the parabola opens upwards, whereas if it is negative, it opens downwards.
Parabolic curves are prominent in real-world scenarios, such as the path of water from a fountain or the trajectory of a ball.
In our original exercise, the equation of the parabolic curve is given by the function \( h = -16 t^2 + 89.6 t \). This equation describes a parabola that opens downwards because the coefficient of \( t^2 \) is negative. This means that its maximum value is its vertex, which represents the highest point of the water above the fountain's main nozzle.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point of the curve, depending on the direction in which the parabola opens. For a parabola opening upwards, the vertex is the lowest point, and for one opening downwards, it is the highest.
The vertex is key for finding extreme values, like maxima or minima, of quadratic functions, such as calculating optimum values in real contexts.
To determine the vertex for the exercise's parabolic function, \( h = -16 t^2 + 89.6 t \), we use the vertex formula: \( t = -b / (2a) \). Substitute \( a = -16 \) and \( b = 89.6 \) to find \( t = -89.6 / (2 \times -16) \). Calculating \( t \), you get the exact moment in time when the water reaches its maximum height.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a form that makes it easier to identify the vertex of the parabola. This form is \( h = a(t - h)^2 + k \), where \( (h, k) \) denotes the vertex.
This process involves rearranging and altering the equation so that it becomes a perfect square trinomial. It is especially useful for solving quadratic equations and understanding the geometry of quadratic functions.
For the equation provided in the exercise, \( h = -16 t^2 + 89.6 t \), you rearrange it into \(-16(t^2 - 5.6t)\). Adding and subtracting the necessary constant completes the square. Although it is more tedious than using the vertex formula directly, it offers a deeper understanding of how parabolas are structured.

Maximum Value of a Quadratic Function

The maximum value of a quadratic function is found at the vertex if the parabola opens downwards. This is because the vertex represents either the peak or trough of the parabolic curve.
In a real-world context, determining the maximum value answers questions like, "What is the highest point?" This is crucial in applications ranging from physics to economics.
In the exercise, once the time \( t \) for the vertex is found using \( t = -b / (2a) \), you substitute it back into the original equation \( h = -16 t^2 + 89.6 t \) to find the maximum height. This substitution yields the highest height reached by the water, representing the function's maximum value. For this example, \( h = -16(t_{max})^2 + 89.6(t_{max}) \), and the result is the maximum height achieved over the fountain.