Chapter 12, Problem 105RE

a.

To determine

Find a model of the form $y=a{x}^3+b{x}^2+cx+d$ .

Answer to Problem 105RE

  $\boxed{y=-3.376068\times {10}^{-7}{x}^3+3.7529\times {10}^{-4}{x}^2-0.17x+132}$

Explanation of Solution

Given information:

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

Use the regression feature of a graphing utility to find a model of the form $y=a{x}^3+b{x}^2+cx+d$

  

  

Calculation:

Consider the given table.

  

To find the regression model use graphing utility $TI-83$

Press start button and observe the screen,

  

Now select EDIT and show display,

  

Now feed the data given in table.

  

Now press start and calc,

  

Now choose cubic regression,

  

Now press ENTER and display will be,

  

Hence the cubic regression model of the above data,

  $\boxed{y=-3.376068\times {10}^{-7}{x}^3+3.7529\times {10}^{-4}{x}^2-0.17x+132}$

b.

To determine

Use the graphing utility to plot the data.

Answer to Problem 105RE

The graphic model is verified.

Explanation of Solution

Given information:

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

  

  

Use the graphing utility to plot the data and graph the model in the same viewing window

Calculation:

Consider the given table.

  

To find the regression model use graphing utility $TI-83$

Press 2^nd and Y= and observe the display,

  

Now press ENTER and show display,

  

Now press WINDOW to choose proper scale.

  

Now press GRAPH and display will be,

  

Now plot the regression equation,

Press Y= and choose plot $2$ using up arrow key. The display will be.

  

Now feed the equation,

  

Now press GRAPH and display will be,

  

Hence the graphic model is verified.

c.

To determine

Find the area of the lot.

Answer to Problem 105RE

Area under the curve is $\boxed{87920}$ square feet.

Explanation of Solution

Given information:

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

  

  

Use the graphing utility to plot the data and graph the model in the same viewing window

Calculation:

Consider the given table.

  

To evaluate the area under the curve,

Define dimension of rectangle,

  $\begin{array}{l}Width\\ \frac{b-a}{n}=\frac{1000-0}{n}\\ =\frac{1000}{n}\\ Height\\ f\left(a+\frac{\left(b-a\right)i}{n}\right)=f\left(0+\frac{\left(1000i\right)}{n}\right)\\ =f\left(\frac{1000i}{n}\right)\\ f\left(x\right)=-3.376068\times {10}^{-7}{x}^3+3.7529\times {10}^{-4}{x}^2-0.17x+132.17\end{array}$

Therefore,

  $\begin{array}{l}f\left(\frac{1000i}{n}\right)=-3.37\times {10}^{-7}{\left(\frac{1000i}{n}\right)}^3+3.75\times {10}^{-4}{\left(\frac{1000i}{n}\right)}^2-0.17\left(\frac{1000i}{n}\right)+132.17\\ =-3.37\frac{i^3}{n^3}+375\frac{i^2}{n^2}-170\frac{i}{n}+132.17\\ \end{array}$

Now approximation of area is sum of areas of all rectangles.

  $\begin{array}{l}A={\displaystyle \sum _{i=1}^{n}f\left(a+\frac{\left(b-a\right)i}{n}\right)}\left(\frac{\left(b-a\right)}{n}\right)\\ A={\displaystyle \sum _{i=1}^n\left(-3.37\frac{i^3}{n^3}+375\frac{i^2}{n^2}-170\frac{i}{n}+132.17\right)}\frac{1000}{n}\\ =1000{\displaystyle \sum _{i=1}^n\left(-3.37\frac{i^3}{n^4}+375\frac{i^2}{n^3}-170\frac{i}{n^2}+\frac{132.17}{n}\right)}\\ =-\frac{337000}{n^4}{\displaystyle \sum _{i=1}^n{i}^3+\frac{375000}{n^3}{\displaystyle \sum _{i=1}^n{i}^2-\frac{170000}{n^2}}{\displaystyle \sum _{i=1}^{n}i+\frac{132170}{n}{\displaystyle \sum _{i=1}^{n}1}}}\end{array}$

Now apply properties of submission,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^n{i}^3={\left[\frac{n\left(n+1\right)}{2}\right]}^2}\\ {\displaystyle \sum _{i=1}^n{i}^2=\left[\frac{n\left(n+1\right)\left(2n+1\right)}{6}\right]}\\ {\displaystyle \sum _{i=1}^{n}i=\left[\frac{n\left(n+1\right)}{2}\right]}\\ {\displaystyle \sum _{i=1}^{n}1=n}\end{array}$

Now calculate the area by taking $\underset{n\to \infty }{\lim }$ .

  $\begin{array}{l}A=\underset{n\to \infty }{\lim }\left\{-84250\left[\frac{n+1}{n}\right]+62500\left[\frac{\left(n+1\right)\left(2n+1\right)}{n^2}\right]-85000\left[\frac{n+1}{n}\right]+132170\right\}\\ =\underset{n\to \infty }{\lim }-84250{\left[1+\frac{1}{n}\right]}^2+62500\left[\left(1+\frac{1}{n}\right)\left(2+\frac{1}{n}\right)\right]-85000\left[1+\frac{1}{n}\right]+132170\\ replace\frac{1}{n}with0\\ A=-84250+125000-85000+132170\\ A=87920\end{array}$

Hence area under the curve is $\boxed{87920}$ square feet.