Chapter 11, Problem 105RE

(a)

To determine

To find: The model of the form $y=a{x}^3+b{x}^2+cx+d$ with the help of regression feature of graphing utility.

Answer to Problem 105RE

The model of the form $y=a{x}^3+b{x}^2+cx+d$ is equal to $y=-0.0000003x^3+0.000375x^2-0.1677x+132.167$ .

Explanation of Solution

Given information:

The table shows the measurements(in feet) of a lot bounded by a stream and two straight roads that meets at right angles(see figure).

$x$	$0$	$100$	$200$	$300$	$400$	$500$	$600$	$700$	$800$	$900$	$1000$
$y$	$125$	$125$	$120$	$112$	$90$	$90$	$95$	$88$	$75$	$35$	0

  

Calculation:

For the regression of the model of the form $y=a{x}^3+b{x}^2+cx+d$ of data follow the steps below:

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now enter the keystrokes $\boxed{\text{STAT}}\boxed{\to }\boxed{6}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(1)

Therefore, the model of the form $y=a{x}^3+b{x}^2+cx+d$ is equal to $y=-0.0000003x^3+0.000375x^2-0.1677x+132.167$ .

(b)

To determine

To graph: The model and data in the same viewing window.

Explanation of Solution

Given information:

The table shows the measurements(in feet) of a lot bounded by a stream and two straight roads that meets at right angles(see figure).

$x$	$0$	$100$	$200$	$300$	$400$	$500$	$600$	$700$	$800$	$900$	$1000$
$y$	$125$	$125$	$120$	$112$	$90$	$90$	$95$	$88$	$75$	$35$	0

  

Graph:

To graph the points on scatter plot, follow the steps using graphing calculator.

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now go to ${\text{Y}}_2$ and enter the right hand side of the function $y=-0.0000003x^3+0.000375x^2-0.1677x+132.167$ . Enter the keystrokes $\boxed{(-)}\boxed{0.0000003}\boxed{\text{X}}\boxed{^}\boxed{3}\boxed{+}\boxed{0.000375}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{-}\boxed{0.1677}\boxed{\text{X}}\boxed{+}\boxed{132.167}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (1).

  

Figure (2)

(c)

To determine

To find: The area of the lot by the model in part(a).

Answer to Problem 105RE

The area of the lot by the model in part(a) is $60033.52{\text{ft}}^2$ .

Explanation of Solution

Given information:

The table shows the measurements(in feet) of a lot bounded by a stream and two straight roads that meets at right angles(see figure).

$x$	$0$	$100$	$200$	$300$	$400$	$500$	$600$	$700$	$800$	$900$	$1000$
$y$	$125$	$125$	$120$	$112$	$90$	$90$	$95$	$88$	$75$	$35$	0

  

Calculation:

As found in part(a), the model for the data is $y=-0.0000003x^3+0.000375x^2-0.1677x+132.167$ .

The area of the curve bounded between $x=0$ and $x=1000$ is given by:

  $\begin{array}{c}{\displaystyle \sum _{i=0}^{1000}y}={\displaystyle \sum _{i=0}^{1000}\left(-0.0000003i^3+0.000375i^2-0.1677i+132.167\right)}\\ ={\displaystyle \sum _{i=0}^{1000}\left(-0.0000003i^3\right)}+{\displaystyle \sum _{i=0}^{1000}\left(0.000375i^2\right)}+{\displaystyle \sum _{i=0}^{1000}\left(-0.1677i\right)}+{\displaystyle \sum _{i=0}^{1000}\left(132.167\right)}\\ =-0.0000003{\displaystyle \sum _{i=0}^{1000}\left(i^3\right)}+0.000375{\displaystyle \sum _{i=0}^{1000}\left(i^2\right)}-0.1677{\displaystyle \sum _{i=0}^{1000}\left(i\right)}+132.167{\displaystyle \sum _{i=0}^{1000}\left(1\right)}\\ =-0.0000003\left(\frac{{\left(1000\right)}^2{\left(1000+1\right)}^2}{4}\right)+0.000375\frac{1000\left(1000+1\right)\left(2\left(1000\right)+1\right)}{6}-0.1677\left(\frac{1000\left(1000+1\right)}{2}\right)+132.167\left(1000\right)\end{array}$

Further simplify,

  $\begin{array}{c}{\displaystyle \sum _{i=0}^{1000}y}=-0.0000003\left(\frac{{\left(1000\right)}^2{\left(1000+1\right)}^2}{4}\right)+0.000375\left(\frac{1000\left(1000+1\right)\left(2\left(1000\right)+1\right)}{6}\right)-0.1677\left(\frac{1000\left(1000+1\right)}{2}\right)+132.167\left(1000\right)\\ =-0.0000003\left(\frac{\left(1000000\right){\left(1001\right)}^2}{4}\right)+0.000375\left(\frac{1000\left(1001\right)\left(2001\right)}{6}\right)-0.1677\left(\frac{1000\left(1001\right)}{2}\right)+132.167\left(1000\right)\\ =-0.0000003\left(\frac{\left(1000000\right)\left(1002001\right)}{4}\right)+0.000375\left(\frac{1000\left(2003001\right)}{6}\right)-0.1677\left(\frac{1001000}{2}\right)+132.167\left(1000\right)\\ =-751.5+12551.87-83933.85+132167\\ =60033.52\end{array}$

Therefore, the area of the lot by the model in part(a) is $60033.52{\text{ft}}^2$ .