Chapter ISG, Problem 9.8.2.4P

a.

To determine

To find: A quadratic function that models the motion of the golf ball and tell what each variable represents.

Answer to Problem 9.8.2.4P

Quadratic function is $y=-16x^2+64x$ , where $x$ represents the time (in seconds) since golf ball is hit and $y$ represents the height of golf ball (in ft).

Explanation of Solution

Given Information: A table containing data-values.

Calculation:

Using quadratic regression feature of the calculator,

  

  $\Rightarrow$ Quadratic function: $y=-16x^2+64x$

Thus, quadratic function is $y=-16x^2+64x$ , where $x$ represents the time (in seconds) since golf ball is hit and $y$ represents the height of golf ball (in ft).

b.

To determine

To explain: The meaning of constant term in the quadratic function in the context of the problem.

Answer to Problem 9.8.2.4P

When time is $0$ , height of golf ball is also $0$ because golf ball has not been hit yet.

Explanation of Solution

Given Information: Function is $y=-16x^2+64x+0$.

  $y=-16x^2+64x+0$

Yes, function has a constant term $0$ which corresponds to the $c-value$ . Value of $c$ simply represents $y-\mathrm{int}ercept$ , in this case, it implies that when time is $0$ height of golf ball is also $0$ because golf ball has not been hit yet.

Thus, when time is $0$ height of golf ball is also $0$ because golf ball has not been hit yet.

c.

To determine

To write: The meaning of coefficients of the quadratic equation.

Answer to Problem 9.8.2.4P

  $a$ implies that first height of golf ball increases over time and then decreases, $b$ affects the maximum height of the golf ball.

Explanation of Solution

Given Information: Function is $y=-16x^2+64x+0$.

In the term $a{x}^2$ , the sign of $a$ determines the direction of the parabola, positive value means that parabola opens upward and negative value signifies that parabola opens downward. In this context it means that first height of golf ball increases over time and then decreases.

In the term $bx$ , $b$ helps in determining axis of symmetry and turning point of a parabola, in this context, it affects the maximum height of the golf ball.

Thus, $a$ implies that first height of golf ball increases over time and then decreases, $b$ affects the maximum height of the golf ball.

d.

To determine

To graph: The function on the coordinate plane at the right, be sure to mark the scale on the axes and label the axes appropriately.

Explanation of Solution

Given Information: Function is $y=-16x^2+64x+0$.

Graph:

  

e.

To determine

To write: An equation to find the times when the golf ball was at a height of exactly $55$ feet and solve the equation, at last explain the steps involved.

Answer to Problem 9.8.2.4P

The golf ball was at a height of $55$ feet when time was $1.25\sec onds\&2.75\sec onds$.

Explanation of Solution

Given Information: Function is $y=-16x^2+64x+0$.

Calculation:

As the height should be equal to $55$ , putting $y=55$ in $y=-16x^2+64x+0$ ,

  $-16x^2+64x=55$

  $\begin{array}{l}\Rightarrow 16x^2-64x+55=0\\ \Rightarrow x=\frac{64\pm \sqrt{{(-64)}^2-4(16)(55)}}{2(16)}\\ =\frac{64\pm \sqrt{576}}{32}\\ =\frac{64\pm 24}{32}\\ =\frac{40}{32},\frac{88}{32}\\ =1.25,2.75\end{array}$

Thus, the golf ball was at a height of $55$ feet when time was $1.25\sec onds\&2.75\sec onds$.

f.

To determine

To find: The maximum height that the golf ball reached and the time at which maximum height was reached.

Answer to Problem 9.8.2.4P

The maximum height of golf ball is $2.64$ ft. at $2$ seconds.

Explanation of Solution

Given Information: Function is $y=-16x^2+64x+0$.

From the graph, the maximum value of $y$ is $2.64$ , i.e., the maximum height is $2.64$ ft.

At this height, the value of $x$ , i.e., the time is $2$ seconds.

Thus, the maximum height of golf ball is $2.64$ ft. at $2$ seconds.