Chapter 5, Problem 8P

a.

To determine

Sketch a scatter plot for the given data set.

Answer to Problem 8P

The scatter plot is

  

Explanation of Solution

Given: A set of the data is,

  $\begin{array}{cccc}Year& Salmon(\times 1000)& Year& Salmon(\times 1000)\\ 1985& 43& 1993& 56\\ 1986& 36& 1994& 63\\ 1987& 27& 1995& 57\\ 1988& 23& 1996& 50\\ 1989& 26& 1997& 44\\ 1990& 33& 1998& 38\\ 1991& 43& 1999& 30\\ 1992& 50& 2000& 22\end{array}$

Calculation:

Let’s take a given data set sketch a scatter plot using MATLAB.

The function is using in the MATLAB to sketch a scatter plot is,

  $\text{scatter(t,y)}$

Program:

clc
clear
close all
t=[1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000];
y=[43 36 27 23 26 33 43 50 56 63 57 50 44 38 30 22];
scatter(t,y,'linewidth',1.25');
set(gca,'Linewidth',1.2,'Fontsize',12);
xlabel('t (Month)');
ylabel('Salmon (x1000)')
axis square
axis tight

Query:

• First, we have defined the given data sets.
• Then using a function “scatter (t, y)” sketch a scatter plot.

b.

To determine

Calculate the cosine function using given data set.

Answer to Problem 8P

The cosine function is,

  $y=20.5\times \cos (0.3927(t-1994))+42.5$

Explanation of Solution

Given: A set of the data is,

  $\begin{array}{cccc}Year& Salmon(\times 1000)& Year& Salmon(\times 1000)\\ 1985& 43& 1993& 56\\ 1986& 36& 1994& 63\\ 1987& 27& 1995& 57\\ 1988& 23& 1996& 50\\ 1989& 26& 1997& 44\\ 1990& 33& 1998& 38\\ 1991& 43& 1999& 30\\ 1992& 50& 2000& 22\end{array}$

Calculation:

First, we have to write a general equation of the cosine function,

  $y=a\cos (w(t-c))+b \mathrm{......}(1)$

Then, calculate the vertical shifting as,

  $b=\frac{1}{2}\left(max value(y)\text{ + }min value(y)\right)=42.5$

Calculate the amplitude as,

  $a=\frac{1}{2}\left(max value(y)\text{ - }min value(y)\right)=20.5$

Then, Calculate the phase shift as,

  $w=\frac{2\pi }{perod}=0.3927$

The value of c is,

  $c=time period at which y value is maximum=1994$

Put all the value into the equation (1) then,

  $y=20.5\times \cos (0.3927(t-1994))+42.5$

Program:

clc
clear
close all
t=[1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000];
y=[43 36 27 23 26 33 43 50 56 63 57 50 44 38 30 22];
b=(1/2)*(max(y)+min(y));
a=(1/2)*(max(y)-min(y));
w=2*pi/16;
idx=find(y==max(y));
c=t(idx);
f=(a*cos(w*(t-c)))+b;

Query:

• First, we have defined the given data sets.
• Then calculate the value of b, a, w, and c.
• Put all the values into the equation of cosine function and get the solution.

c.

To determine

Sketch a graph of the function which is found in part (b).

Answer to Problem 8P

The solution is,

  

Explanation of Solution

Given: A set of the data is,

  $\begin{array}{cccc}Year& Salmon(\times 1000)& Year& Salmon(\times 1000)\\ 1985& 43& 1993& 56\\ 1986& 36& 1994& 63\\ 1987& 27& 1995& 57\\ 1988& 23& 1996& 50\\ 1989& 26& 1997& 44\\ 1990& 33& 1998& 38\\ 1991& 43& 1999& 30\\ 1992& 50& 2000& 22\end{array}$

Calculation:

Sketch a graph of the cosine function in MATLAB using function “plot (f, t)”.

The function is found in part (b) is,

  $y=20.5\times \cos (0.3927(t-1994))+42.5$

Program:

clc
clear
close all
t=[1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000];
y=[43 36 27 23 26 33 43 50 56 63 57 50 44 38 30 22];
b=(1/2)*(max(y)+min(y));
a=(1/2)*(max(y)-min(y));
w=2*pi/16;
idx=find(y==max(y));
c=t(idx);
f=(a*cos(w*(t-c)))+b;
scatter(t,y,'linewidth',1.25');
hold on
plot(t,f,'linewidth',1.25');
set(gca,'Linewidth',1.2,'Fontsize',12);
xlabel('t (Month)');
ylabel('Salmon (x1000)')
axis square
axis tight

Query:

• First, we have defined the given data sets.
• Then calculate the value of b, a, w, and c.
• Put all the values into the equation of cosine function and get the solution.
• Then sketch a graph.

d.

To determine

Calculate the sine function using given data set.

Answer to Problem 8P

The cosine function is,

  $y=20.5\times \sin (0.3927(t+1994))+42.5$

And the best fitting curve is,

  

Explanation of Solution

Given: A set of the data is,

  $\begin{array}{cccc}Year& Salmon(\times 1000)& Year& Salmon(\times 1000)\\ 1985& 43& 1993& 56\\ 1986& 36& 1994& 63\\ 1987& 27& 1995& 57\\ 1988& 23& 1996& 50\\ 1989& 26& 1997& 44\\ 1990& 33& 1998& 38\\ 1991& 43& 1999& 30\\ 1992& 50& 2000& 22\end{array}$

Calculation:

First, we have to write a general equation of the cosine function,

  $y=a\sin (w(t-c))+b \mathrm{......}(1)$

Then, calculate the vertical shifting as,

  $b=\frac{1}{2}\left(max value(y)\text{ + }min value(y)\right)=42.5$

Calculate the amplitude as,

  $a=\frac{1}{2}\left(max value(y)\text{ - }min value(y)\right)=20.5$

Then, Calculate the phase shift as,

  $w=\frac{2\pi }{perod}=0.3927$

The value of c is,

  $c=time period at which y value is maximum=1994$

Put all the value into the equation (1) then,

  $y=22.9\times \sin (0.5236(t+7))+62.9$

Program:

clc
clear
close all
t=[1 2 3 4 5 6 7 8 9 10 11 12];
y=[40.0 43.1 54.6 64.2 73.8 81.8 85.8 83.9 76.9 66.8 55.5 44.5];
b=(1/2)*(max(y)+min(y));
a=(1/2)*(max(y)-min(y));
w=2*pi/max(t);
idx=find(y==max(y));
c=t(idx);
f=(a*sin(w*(t+c)))+b;
scatter(t,y,'linewidth',1.25');
hold on
plot(t,f,'linewidth',1.25');
set(gca,'Linewidth',1.2,'Fontsize',12);
xlabel('t (Month)');
ylabel('Average temperature (^{\circ}F)')
axis square
axis tight

Query:

• First, we have defined the given data sets.
• Then calculate the value of b, a, w, and c.
• Put all the values into the equation of cosine function and get the solution.
• Then sketch a best fitting curve with the scatter plot.