Question

A water tank consists of a cylindrical part of radius r and height h and a hemispherical top. The tank is to be constructed to hold 600 m³ when filled. The surface area of the cylindrical part is 2πrh, and its volume is πr²h. The surface area of the hemispherical top is given by 2πr², and its volume is given by 2πr³/3. The cost to construct the cylindrical part of the tank is $400 per square meter of surface area; the hemispherical part costs $600 per square meter. Use the fminbnd function to compute the radius that results in the least cost. Compute the corresponding height h.

 

 

Answer 1

Program plan:

• Write a function having two arguments height and the total cost of constructions as the function of radius.

• ‘fminbnd’ is used to find the least value of the function between the given intervals.

 

Program:

%**********************************************************

%A function is declared for the total cost and height as

%the arguments and the radius as the variable, fminbnd is

%used to find the radius of the least cost.

%**********************************************************

%function of radius

function [totalcost,height]=tc(r)

%‘height’ and ‘totatcost’ are the variables declared in the function.

height=600./(pi.*r.^2)-2.*r./3;

totalcost=2.*pi.*r.*height.*400+2.*pi.*600.*r.^2;

end

%main program

%‘fminbnd’ is used to find the minimum value of the function in the given interval.

rmin=fminbnd(@tc,0,10);

%‘rmin’ is assigned with the radius corresponding to the minimum total cost

disp('The radius which has least total cost is ');

%displays rmin value

disp(rmin);

%This line is used to call the function and assign the values of total cost and height to ‘[t,h]’

[t,h]=tc(rmin);

%displays 'Total minimum cost' text

disp('Total minimum cost');

%displays t variable value

disp(t);

%displays text

disp('The corresponding height is ');

%displays h variable value

disp(h);

 

Output:

The radius which has least total cost is 4.8572.

Total minimum cost 1.4823e+05.

The corresponding height is 4.8572.