Chapter 3, Problem 2P

(a)

To determine

To find: The quadratic polynomial that best fits the given data.

Answer to Problem 2P

The quadratic polynomial that best fits the given data is equal to $y=-0.2783x^2+18.4654x-166.7321$ .

Explanation of Solution

Given information:

The more corn a farmer plants per acre, the greater is the yield the farmer can expect, but only up to a point. Too many plants per acre can cause overcrowding and decrease yields. The data give crop yields per acre for various densities of corn plantings, as found by researchers at a university test farm.

Density(plants/acre)	Crop yield(bushels/acre)
$15000$	$43$
$20000$	$98$
$25000$	$118$
$30000$	$140$
$35000$	$142$
$40000$	$122$
$45000$	$93$
$50000$	$67$

Calculation:

Take the density of the plants per thousand i.e. $x=1$ represents $1000$ plants per acre.

For the quadratic model 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 $\boxed{\text{STAT}}\boxed{\to }\boxed{5}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(1)

Therefore, the quadratic polynomial that best fits the given data is equal to $y=-0.2783x^2+18.4654x-166.7321$ .

(b)

To determine

To graph: The scatter plot of the given data and the quadratic polynomial obtained from part(a) in the same viewing window.

Explanation of Solution

Given information:

The more corn a farmer plants per acre, the greater is the yield the farmer can expect, but only up to a point. Too many plants per acre can cause overcrowding and decrease yields. The data give crop yields per acre for various densities of corn plantings, as found by researchers at a university test farm.

Density(plants/acre)	Crop yield(bushels/acre)
$15000$	$43$
$20000$	$98$
$25000$	$118$
$30000$	$140$
$35000$	$142$
$40000$	$122$
$45000$	$93$
$50000$	$67$

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 press the key $\boxed{\text{Y}=}$ and go to ${\text{Y}}_2$ and enter the right hand side of the function $y=-0.2783x^2+18.4654x-166.7321$ . Enter the keystrokes $\boxed{(-)}\boxed{0.2783}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{18.4654}\boxed{\text{X}}\boxed{-}\boxed{166.7321}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (2).

  

Figure (2)

(c)

To determine

To find: The yield for $37000$ plants per acre.

Answer to Problem 2P

The yield for $37000$ plants per acre is equal to $135.49$ bushels/acre.

Explanation of Solution

Given information:

The more corn a farmer plants per acre, the greater is the yield the farmer can expect, but only up to a point. Too many plants per acre can cause overcrowding and decrease yields. The data give crop yields per acre for various densities of corn plantings, as found by researchers at a university test farm.

Density(plants/acre)	Crop yield(bushels/acre)
$15000$	$43$
$20000$	$98$
$25000$	$118$
$30000$	$140$
$35000$	$142$
$40000$	$122$
$45000$	$93$
$50000$	$67$

Calculation:

As observed from the graph of the function in part(b), the value of the function at $x=37$ is equal to $135.49$ .

So, the crop yield for $37000$ plants/acre is equal to $135.49$ bushels/acre.

Therefore, the yield for $37000$ plants per acre is equal to $135.49$ bushels/acre.