Chapter 1.8, Problem 73PS

Solution

a)

To make: A table of values giving the position $\left(x,y\right)$ of the basketball after $0,0.25,0.5,0.75$ , and $1$ second

Table is as follows.

  $\begin{array}{cccc}t& x& y& \left(x,y\right)\\ 0& 0& 6& \left(0,6\right)\\ 0.25& 5& 10.25& \left(5,10.25\right)\\ 0.5& 10& 12.5& \left(10,12.5\right)\\ 0.75& 51& 12.75& \left(51,12.75\right)\\ 1& 20& 11& \left(20,11\right)\end{array}$

Given:

The path of the basketball can be modeled by the parametric equations $x=20t,y=-16t^2+21t+6$ where $x$ and $y$ are measured in feet, $t$ is measured in seconds, and the player's feet are at $\left(0,0\right)$ .

Calculation:

Substitute $0$ for $t$ in $x=20t$ and solve for $x$ .

  $\begin{array}{c}x=20\left(0\right)\\ =0\end{array}$

Substitute $0$ for $t$ in $y=-16t^2+21t+6$ and solve for $y$ .

  $\begin{array}{c}y=-16{\left(0\right)}^2+21\left(0\right)+6\\ =6\end{array}$

When $t=0$ , the point $\left(x,y\right)$ is at $\left(0,6\right)$ .

Similarly, we can find $\left(x,y\right)$ for the remaining given values of $t$ .

Organize the results in a table.

  $\begin{array}{cccc}t& x& y& \left(x,y\right)\\ 0& 0& 6& \left(0,6\right)\\ 0.25& 5& 10.25& \left(5,10.25\right)\\ 0.5& 10& 12.5& \left(10,12.5\right)\\ 0.75& 51& 12.75& \left(51,12.75\right)\\ 1& 20& 11& \left(20,11\right)\end{array}$

b)

To graph: The parametric equations.

Graph is as follows.

  

Given:

The path of the basketball can be modeled by the parametric equations $x=20t,y=-16t^2+21t+6$ where $x$ and $y$ are measured in feet, $t$ is measured in seconds, and the player's feet are at $\left(0,0\right)$ .

Graph:

Plot the points $\left(0,6\right),\left(5,10.25\right),\left(10,12.5\right),\left(51,12.75\right),\left(20,11\right)$ on a coordinate plane.

  

Join the points with a smooth curve.

  

c)

To solve: The position of the basketball rim is $\left(15,10\right)$ . The top of the backboard is $\left(15,12\right)$ . Does the player make the free throw?

The player does not make the free throw as the height of the ball is above the backboard.

Given:

The path of the basketball can be modeled by the parametric equations $x=20t,y=-16t^2+21t+6$ where $x$ and $y$ are measured in feet, $t$ is measured in seconds, and the player's feet are at $\left(0,0\right)$ .

Calculation:

From the table obtained in part $\left(a\right)$ , the $y$ -value is $12.75\text{ft}$ when the value of $x$ is $15$ . It should be $10\text{ft}$ to make a free throw.

Since the height of the ball is above the backboard, the player does not make the free throw.