Chapter 1.7, Problem 16GP

Solution

To find: The maximum height of the baseball whose height is given by the function $y=-16t^2+80t+2$ .

The maximum height of the baseball is $102$ .

Given information:

The height of the baseball is given by the function $y=-16t^2+80t+2$ .

Concept used:

Vertex form a quadratic function is given as $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the quadratic function.

The vertex of the quadratic function represents the maximum/minimum point of the function.

If $a$ positive, the vertex is the minimum point and if $a$ is negative, the vertex is the maximum point of the function.

Calculation:

Find the vertex of the given function by writing it into vertex form.

  $\begin{array}{c}y=-16t^2+80t+2\\ =-16\left(t^2-5t\right)+2\\ =-16\left(t^2-2\cdot x\cdot \frac{5}{2}+{\left(\frac{5}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2\right)+2\\ =-16\left({\left(t-\frac{5}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2\right)+2\\ =-16{\left(t-\frac{5}{2}\right)}^2+16{\left(\frac{5}{2}\right)}^2+2\\ =-16{\left(t-\frac{5}{2}\right)}^2+16\cdot \frac{25}{4}+2\\ =-16{\left(t-\frac{5}{2}\right)}^2+100+2\\ =-16{\left(t-\frac{5}{2}\right)}^2+102\end{array}$

Here, $a=-16$ , which is negative. So, the vertex of the function is the maximum point of the function.

The vertex is $\left(\frac{5}{2},102\right)$ . Therefore, the maximum height of the baseball is $102$ .

Conclusion:

The maximum height of the baseball is $102$ .