Chapter 9.3, Problem 37PPS

a.

To determine

To find the duration of time that “S” is in the air.

Answer to Problem 37PPS

The duration of time is 2.2 seconds .

Explanation of Solution

Given information :

The function provided for the path of the ball is $h=-16t^2+30t+10$ .

Formula used :

The given equation would be graphed as a function and the points of x-intercepts would be the results. For these points, the x-coordinate shall serve the purpose of the results.

Since the roots are not on points that are integers, use the approximation method to compute the roots.

Calculation :

Using the graphing calculator, the graph for the given function is:

In this, the parabola one x-intercept in between 2 and 3.

To find the roots, make a table of values for tin between these points with intervals on 0.1 . The value of x that is result closest to 0 will be the roots of the equation.

Table of values in between 2 and 3

X	2.1	2.2	2.3	2.4	2.5	2.6	2.7	2.8	2.9
Y	2.44	-1.44	-5.64	-10.16	-15	-20.16	25.64	-31.44	-37.56

In the tables, the value of y that is closest to 0 is -1.44. Their corresponding x values would be the roots.

Thus, 2.2 is the root. Hence the time taken to reach the ground is 2.2 seconds .

b.

To determine

To calculate the time when “S” reaches 15 feet.

Answer to Problem 37PPS

The time at which “S” reaches 15 feet are 0.2 seconds and1.7 seconds .

Explanation of Solution

Given information :

The function provided is $h=-16t^2+30t+10$ and $h=15$ feet.

So the quadratic equation would be:

  $15=-16t^2+30t+10$

Replacing the value of h

  $or,-16t^2+30t-5=0$

Formula used :

Since h is given, put the value of h in the given function and calculate t . This would give the time needed to reach the given height.

Calculation :

Using the graphing calculator, the graph for the given function is:

In this, the parabola one x-intercept in between 1 and 2 and the other one between 0 and 1. These would be the two time frames when the height of “S” would be at 15 feet.

To find the roots, make a table of values for t in between these points with intervals on 0.1 . The value of x that is result closest to 0 will be the roots of the equation.

Table of values in between 0 and 1

X	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
Y	-5.76	-2.84	-0.24	2.04	4	5.64	6.96	7.96	8.64

Table of values in between 2 and 3

X	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
Y	8.64	7.96	6.96	5.64	4	2.04	-0.24	-2.84	-5.76

In the tables, the value of y that is closest to 0 is -0.24. Their corresponding x values would be the roots.

Thus, 0.2 and 1.7are the roots. Hence the time at which she is 15 feet above the ground are at0.2 seconds and 1.7 seconds .

c.

To determine

To calculate whether “S” was able to score the extra points.

Answer to Problem 37PPS

The maximum height “S” reaches is 24.0625 feet. Hence she will score the extra points.

Explanation of Solution

Given information :

The function provided is $h=-16t^2+30t+10$ .

Formula used :

The maximum height attained can be found by the y-intercept of the vertex. Compare this with 20 feet TO-DETERMINE whether the extra points were scored or not.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ . Here, $a=-16$ and $b=76$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. Use this value in the original function to derive the maximum distance travelled.

Calculation :

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{30}{2\times \left(-16\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{30}{32}$

Simplifying this

  $x=\frac{15}{16}or0.9375$ is the equation of axis of symmetry.

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y . These two coordinates of x and y would be the point where the vertex is.

  $=-16{\left(0.9375\right)}^2+30\left(0.9375\right)+10$

Putting the value of $x=0.9375$ in the original function.

  $=-14.0625+28.125+10$

Simplifying the expression.

  $=24.0625$

Thus, the maximum distance of the jump is 24.0625 feet. This is greater than 20 feet and hence, the extra points will be scored.