Chapter 1, Problem 21TP

Solution

a.

To draw: A scatter plot of the data.

Given information:

The data is given below.

  

Calculation:

Plot the scatter plot based on the given data:

  

b.

To draw: The parabola the parabola that best the data.

Given information:

The data is given below.

  

Draw the curve of best fits:

Based on the data draw a parabola that best fits the given data.

The graph is given below.

  

c.

To write: A quadratic model for the data by using the coordinates points on the parabola.

  $y=0.4x^2+2.7x+21$ is the quadratic model for the given data.

Given information:

The parabola is given below.

  

Concept used:

The standard form of the parabola is $y=a{x}^2+bx+c$ .

Calculations:

The standard form of the parabola is $y=a{x}^2+bx+c$ .

Now take the coordinate $\left(0,21\right)$ from the parabola and these coordinate passing through the parabola, the equation becomes:

  $\begin{array}{c}21=0\times a+0\times b+c\\ c=21\end{array}$

Now consider the coordinate point $\left(2,28\right)$ ,which passing through the parabola:

Then $$

  $\begin{array}{l}4a+2b=28-21\\ 4a+2b=7\end{array}$

Also consider another coordinate point $\left(10,88\right)$ , which passing through the parabola:

Then $$

  $\begin{array}{l}100a+10b=88-21\\ 100a+10b=67\end{array}$

Now the by solve the system of equations;

  $\begin{array}{l}4a+2b=7\left(1\right)\\ 100a+10b=67\left(2\right)\\ \left(1\right)\times 5=20a+10b=35\left(3\right)\\ \left(2\right)-\left(3\right)\Rightarrow 80a=32\\ a=0.4\\ \text{Substitute}\text{}\text{}\text{}a=0.4\text{in}\left(1\right)\\ 2b=7-1.6\\ b=2.7\end{array}$

Now substitute the values of the coefficient into the equation of parabola then

  $y=0.4x^2+2.7x+21$ is the quadratic model for the given data.

d.

To make: The table of given data by using model obtained in part (c).

Given information:

The parabola is equation: $y=0.4x^2+2.7x+21$

Calculations:

Now make the table for same value of $x$ in the original table.

  $\begin{array}{cc}x& y\\ 0& 21\\ 1& 24\\ 2& 28\\ 3& 33\\ 4& 38\\ 5& 45\\ 6& 52\\ 7& 60\\ 8& 68\\ 9& 78\\ 10& 88\\ 11& 99\\ 12& 111\\ 13& 124\\ 14& 137\\ 15& 152\end{array}$

Now compare the obtained value with given data.

  $\begin{array}{cc}x& y\\ 0& 21\\ 1& 24\\ 2& 28\\ 3& 33\\ 4& 38\\ 5& 45\\ 6& 52\\ 7& 60\\ 8& 68\\ 9& 78\\ 10& 88\\ 11& 99\\ 12& 111\\ 13& 124\\ 14& 137\\ 15& 152\end{array}\begin{array}{c}\text{Original}\\ 21\\ 21\\ 28\\ 36\\ 41\\ 45\\ 47\\ 52\\ 59\\ 63\\ 88\\ 93\\ 112\\ 125\\ 153\\ 184\end{array}$