Chapter 1.7, Problem 39E

Solution

To find: The vertex of the function’s graph and to tell what the vertex represents.

The vertex of the given function's graph is $\left(2.8,125.44\right)$ .

The water will reach a maximum height of $125.44$ feet at $2.8$ seconds.

Given information:

At Buckingham Fountain in Chicago, the water's height $h$ (in feet) above the main nozzle can be modeled by $h=-16t^2+89.6t$ where $t$ is the time (in seconds) since the water has left the nozzle.

  

Calculation:

First, factor out $-16$ from the right side.

  $h=-16\left(t^2-5.6t\right)$

Square half the coefficient of $t$ .

  $\begin{array}{c}{\left(-\frac{5.6}{2}\right)}^2={\left(-2.8\right)}^2\\ =7.84\end{array}$

Now, complete the square. For this, subtract $16\left(7.84\right)$ from each side of the equation.

  $\begin{array}{c}h-16\left(7.84\right)=-16\left(t^2-5.6t\right)-16\left(7.84\right)\\ h-125.44=-16\left(t^2-5.6t+7.84\right)\end{array}$

Write $t^2-5.6t+7.84$ as a binomial squared.

  $h-125.44=-16{\left(t-2.8\right)}^2$

Solve for $h$ . For this, add $125.44$ to each side.

  $\begin{array}{c}h-125.44+125.44=-16{\left(t-2.8\right)}^2+125.44\\ h=-16{\left(t-2.8\right)}^2+125.44\end{array}$

A quadratic function in the form $y=a{\left(x-h\right)}^2+k$ is said to be in vertex form, where $\left(h,k\right)$ is the vertex.

On comparing the equation obtained with the vertex form, $a=-16,h=2.8$ , and $k=125.44$ . Thus, the vertex of the given function's graph is $\left(2.8,125.44\right)$ .

The $y-$ coordinate of the vertex of a graph represents the maximum or minimum value of the function.

Since $a<0$ , the parabola of the given function opens down and it has the maximum value. Therefore, the water will reach a maximum height of $125.44$ feet at $2.8$ seconds.