Chapter 9.1, Problem 58PPS

(a)

To determine

To Graph:

The equation.

Explanation of Solution

Given Information:

Miranda has her boat docked on the west side of Casper Point.She is boating over to the Casper Marina.The distance travelled by Miranda over time can be modeled by the equation $d=-16t^2+66t$ ,where d is the number of feet she travels in t minutes.

  

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.
• Parent Function:

  $y=x^2$

• Standard Form :

  $f(x)=a{x}^2+bx+c;a\ne 0$

• Type of Graph:

Parabola

• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

  $x=-\frac{b}{2a}$

• y-intercept = The y coordinate of the point at which the graph intersects y-axis = c
• Graph:

  

• 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 $d=-16t^2+66t$ ; comparing it with standard form of quadratic equation , we have a = -16 , b = 66 , c = 0 .
• Find equation of Axis of Symmetry :

  $\begin{array}{l}d=a{t}^2+bt+c\mathrm{...........................}[s\tan dard\_form]\\ t=-\frac{b}{2a}\mathrm{.....................................}[formula\_of\_axis\_of\_symmetry]\\ t=-\frac{66}{2(-16)}=\frac{33}{16}\approx \mathrm{2..........................}[a=-16,b=66]\end{array}$

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

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

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

Substituting the value of $t=2$in the equation $d=-16t^2+66t$

  $\begin{array}{l}d=-16t^2+66t\\ $ d=-16{(2)}^2+66(2)\mathrm{...............}[t=2]$d=-64+\mathrm{132.........................}[simplify]\\ d=68\end{array}$

The vertex is at $(2,68)$ .

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

• Find d-intercept :

The d-coordinate of the point at which the graph cuts the d-axis is the d-intercept.

d-intercept always occurs at (0,c) and c = 0 here.

So, d-intercept = 0 and the point is (0,0)

• The axis of symmetry cuts the parabola into two equal parts . So, if there is a point on one side, there is a corresponding point on the other side that is at a same distance from the axis of symmetry and the same d-value.

t	d
1	50
2	68
3	50

• Connect the points with a smooth curve.

Graph :

(b)

To determine

To Find:

The maximum number of feet north that she travelled.

Answer to Problem 58PPS

68 feet

Explanation of Solution

Given Information:

Miranda has her boat docked on the west side of Casper Point.She is boating over to the Casper Marina.The distance travelled by Miranda over time can be modeled by the equation $d=-16t^2+66t$ ,where d is the number of feet she travels in t minutes.

  

From part (a) , the vertex is (2,68) and it is the maximum value of the graph.

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.

• y-intercept = The y coordinate of the point at which the graph intersects y-axis = c
• 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 $d=-16t^2+66t$ ; comparing it with standard form of quadratic equation , we have a = -16 , b = 66 , c = 0 .
• To find the maximum number of feet that Miranda travelled:

Since, a = -16 < 0 . So, the graph of the function opens downward and the vertex is the maximum value.

The y-coordinate of the vertex is the maximum number of feet that Miranda travelled .

Vertex is (2,68) . So, the maximum number of feet that Miranda travelled is 68 feet.

(c)

To determine

To Find:

Time taken by Miranda to reach Casper Marina.

Answer to Problem 58PPS

4 minutes

Explanation of Solution

Given Information:

From part (a) , we have the vertex is $(2,68)$ and it is the maximum of the function.

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.
• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

Calculation:

• The vertex is (2,68)
• It takes the boat 2 minutes to reach the vertex, and due to symmetry it will take the boat another 2 minutes to reach Casper Marina.
• So, it will take Miranda 2 + 2 = 4 minutes to reach Casper Marina.