Chapter 12, Problem 105RE

a.

To determine

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

Answer to Problem 105RE

  $y=-3.37607\times {10}^{-7}{x}^3+0.000375291x^2-0.16772x+\mathrm{132.168.}$

Explanation of Solution

Given information: The table shows the measurement (in feet) of a plot of land bounded

by a stream and two straight roads that meet at right angles .

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:

Using the graphing utility on above table data to find the model:

  $\begin{array}{c}\text{polynomials}\text{of}\text{degree}3\\ y=a{x}^3+b{x}^2+cx+d\\ a=-3.37607\times {10}^{-7}.\\ b=\mathrm{0.000375291.}\\ c=-\mathrm{0.16772.}\\ d=\mathrm{132.168.}\end{array}$

  $y=-3.37607\times {10}^{-7}{x}^3+0.000375291x^2-0.16772x+\mathrm{132.168.}$

b.

To determine

To graph: the model by using graphing utility to plot the data.

Answer to Problem 105RE

Explanation of Solution

Given information: The table shows the measurement (in feet) of a plot of land bounded

by a stream and two straight roads that meet at right angles .

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:

The graph of the model on the above is given below.

  

c.

To determine

To estimate: the area of the plot of land by using the model in part (a).

Answer to Problem 105RE

The area is $A\approx 87695squarefeet.$

Explanation of Solution

Given information: The table shows the measurement (in feet) of a plot of land bounded

by a stream and two straight roads that meet at right angles .

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:

From the given data a=0 ,b $\approx$ 1000.

Each rectangle has a width of $\frac{b-a}{n}=\frac{1000}{n}.$

Using right endpoints ,each height is $f(a+\frac{(b-a)}{n}i)=f(\frac{1000i}{n}).$

  $\begin{array}{l}A\approx {\displaystyle {\sum }_{i=1}^{n}f(\frac{1000i}{n}})\frac{1000}{n}.\\ A\approx {\displaystyle {\sum }_{i=1}^n[(-3.376068\times {10}^{-7})(}\frac{1000i}{n}{)}^3+(3.7529\times {10}^{-4}){(}^{\frac{1000i}{n}}-0.17(\frac{1000i}{n})+132]\frac{1000}{n}\\ A\approx {\displaystyle {\sum }_{i=1}^1[\frac{-337606.8i^3}{n^4}}+\frac{375290i^2}{n^3}-\frac{170000i}{n^2}+\frac{132000}{n}]\\ A\approx -\frac{84401.7{(n+1)}^2}{n^2}+\frac{187645(n+1)(2n+1)}{3n^2}-\frac{85000(n+1)}{n}+132000.\\ \text{The}\text{exact}\text{area}\text{is}\text{:}\\ A\approx \underset{n\to \infty }{\lim }[-\frac{84401.7{(n+1)}^2}{n^2}+\frac{187645(n+1)(2n+1)}{3n^2}-\frac{85000(n+1)}{n}+132000].\\ A=-84401.7-0-0+\frac{375290}{3}+0+0-85000-0+132000.\\ A\approx 87695squarefeet.\end{array}$