Chapter 9.1, Problem 6CCYP

To determine

To Find:

The maximum height of the javelin.

Answer to Problem 6CCYP

70 feet

Explanation of Solution

Given Information:

Emilio is competing in the javelin throw.The height of the javelin can be modeled by the equation $y=-16x^2+64x+6$ ,where y represents the height in feet of the javelin after x seconds.

Concept Used :

• Quadratic Functions:
• Quadratic functions are non-linear and are of the form:

  $f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

• Shape of the graph of a Quadratic function is called Parabola.
• Parabolas are symmetric about a central line called Axis Of Symmetry.
• The axis of symmetry intersects a parabola only at one point , called the Vertex.
• If a > 0 ,
• The graph of $a{x}^2+bx+c$ opens upward.
• The lowest point on the graph is the minimum.
• If a < 0 ,
• The graph of $a{x}^2+bx+c$ opens downward.
• The highest point on the graph is the maximum.
• The maximum or the minimum is the vertex.

Calculation:

• In the equation $y=-16x^2+64x+6$ ; comparing it with standard form of quadratic equation , we have a = -16 , b = 64 , c = 6 .
• Find equation of Axis of Symmetry :

  $\begin{array}{l}y=a{x}^2+bx+c\mathrm{...........................}[s\tan dard\_form]\\ x=-\frac{b}{2a}\mathrm{.....................................}[formula\_of\_axis\_of\_symmetry]\\ x=-\frac{64}{2(-16)}=\mathrm{2..........................}[a=-16,b=64]\end{array}$

The equation of axis of symmetry is $x=2$

• Find the Vertex and determine whether it is maximum or minimum:

Since axis of symmetry passes through the vertex , we have x coordinate of the vertex (x,y) is $x=2$

Substituting the value of $x=2$in the equation $y=-16x^2+64x+6$

  $\begin{array}{l}y=-16x^2+64x+6\\ $ y=-16{(2)}^2+64(2)+\mathrm{6...............}[x=2]$y=-64+2(64)+\mathrm{6..................................}[simplify]\\ y=70\end{array}$

The vertex is at $(2,70)$ .

Since the a = -16 < 0 .So, the graph opens downward .Hence, the vertex is the maximum.

• The maximum height occurs at vertex.
• So, the maximum height the javelin reaches is 70 feet after 2 seconds of release.