Question

The Depth Gauge Problem Liquids are often stored in elliptical storage tanks as shown below. To measure the volume of liquid in the tank, a depth gauge (measuring stick) can be used. It is inserted into an opening at the top and the level of liquid on the gauge can be used to determine the amount of liquid in the tank.

The tank has width w, height h and length len (all in meters). In the example output shown below, we take w=8, h=4 and len=7. Your programs should work for any values of w, h and len, not just these specific values.

The assignment is divided into two parts.

Part 1:

In the first part of the assignment we look at inserting a measuring stick that is already calibrated in units of 10 centimeters. This measuring gauge can be inserted into an opening at the top of the tank and used to measure the depth of the liquid in the tank.

Your task will be to write a C program to produce a table of values showing the volume of liquid in the tank for each of the points on the gauge.

The output of your program (for the example above) should look like:

Depth 10 cm : Volume 1.188814 cubic meters

Depth 20 cm : Volume 3.336448 cubic meters

Depth 30 cm : Volume 5.992683 cubic meters

. . .

Depth 380 cm : Volume 172.547399 cubic meters

Depth 390 cm : Volume 174.657114 cubic meters

Depth 400 cm : Volume 175.743037 cubic meters

Methodology for Part 1:

If the tank has width W and height H (in centimeters), the focal radii of the cross section are A = W/2 and B = H/2. Then the equation of the ellipse is:

X2/A2 + Y2/B2 = 1

To find the volume at given depth you should compute the cross-sectional area of the tank for each given depth using a numerical integration algorithm such as the trapezoidal method. You must use a general integration function and apply it to this particular function. Do not write an integration function that is specific to this problem. Do not use an analytic solution to integrate the function. Then multiply this by the length of the tank.

Hint: It is probably easier to imagine the tank on its side so that the depth gauge is inserted horizontally. If you do this you must express the equation as a function of y and integrate that function.

Part 2:

The second part of the assignment is a variation on this. In this case the stick is not calibrated. We would like to calibrate it. Rather than calibrate it by equidistant markings, we calibrate it to show at what level the tank contains a certain volume of the liquid.

You are to write a C program that determines where the gauge should be marked (to the nearest millimeter) corresponding to volumes of 5, 10, 15, … cubic meters (up to the total volume of the tank).

The output of your program (for the example above) should look like:

Volume 5: Depth 26.54 cm

Volume 10: Depth 42.48 cm

Volume 15: Depth 56.08 cm

. . .

Volume 165: Depth 355.12 cm

Volume 170: Depth 370.52 cm

Volume 175: Depth 392.03 cm

Methodology for Part 2:

If we view the volume as a function of the position y, this problem reduces to the problem of finding a root of the equation V(y) = depth, for the various depths. While the function V(y) is not given by a simple formula, it can be determined for any y by using the numerical integration as defined in part 1. Write a program to determine the roots of this equation for each depth by using a root finding algorithm such as the bisection method. Again, you must use a general root finding function and apply it to this particular function. Do not use an analytic solution to solve this problem. Each function evaluation will have to be done using an integration algorithm such as the trapezoidal method.

Answer 1

Answer:

Program code screen shot:

CRT_SECURE_NO WARNINGS #define #include <stdio.h> #include <math.h> #include <stdlib.h> // define the global variable double width; double height double length; //declare a function function ) to compute the // cross section area of the tank double function (double x); /declare the function num Integration trape zoidal () / to compute the intergration value by using trapezoidal method double num Integration trapezoidal (double A, double B, int n) // define the main function int main () /define a variable to keep maintining the calibrated /units of the stick int calibratedUnit 0 ; define a variable to hold the upper bound of the /integration value int upperBound = 100; /declare the required variables double stickHeight, area_of_tank, volume_of_tank; printf ("To compute the volume at the given depth at" "certain cross sectional area\n\n"); / prompt the user for the width, height and length of the / tank in meters printf ("Enter the data measurements in meters\n"); /prompt and read the width of the tank printf ("Enter tank width (m) scanf ("$1f", &width) ") ; /prompt and read the height of the tank printf ("Enter tank height (m) scanf ("1f", aheight)) "); prompt and read the length of th printf("Enter tank length (m): ") ; scanf ("1f", length); tank

at every calibratedUnit, find the area and volume and display the //volume cubic meters height 100; calibratedUnit = 10) 10 calibratedUnit <= for (calibratedUnit 1find the stick height stickHeight calibratedUnit / 100.0; = /call the num_Integration_trapezoidal () function /to find the area area_of_tank = num_Integration_trapezoidal (heighht/2.0-stickHeight, height /2.0, upperBound): / compute the volume of the tank volume of tank = area of tank * length; /display the depth and volume of the tank printf ("Depth id cm: Volume .61f cubic metres\n", calibratedUnit, volume of tank) ; // to halt the program ; // to halt the program get char get char return 0: // declare a function function ) that accepts a double value // and returns a double // This function is to compute the cross-section area of the tank double function (double x) double A = height 2.0 double B= width / 2.0; double y = (B / A) * sgrt (A * A - x * x) return y

// define a function num_Integration_trapezoidal that accepts // two double values and an integer value and returns a double value. // This function is to find the interqrated value of the cross-section area /by using trapezoidal method double num Integration trapezoidal (double A, double B, int n) /declare the required variables double xVal, dx, totalSum = 0.0; = 0; double result int i 0 /find the interval value of the integration dx (B - A) / n; / find the initial totalSum = (function (A) sum value function (B) ) /2; /add the value A with interval xVal A + dx; /loop through each value of the interval till it reaches the /upper bound for (i= 1; i<n; i++) /find the sum of the cross-section area totalSum += function (xVal); /increment the interval value xVal +dx; /find the total area result = 2 * totalSum * dx: /return the result return result:

Sample Output:

To compute the volume at the given depth at certain cross sectional area Enter the data measurements in meters Enter tank width<m: 8 Enter tank height <m>: 4 Enter tank length(m): 7 Depth 10 cm: Depth 20 cm: Depth 30 cm: Depth 40 cm: Depth 50 cm: Depth 60 cm: Depth 70 cm: Depth 80 cm: Depth 90 cm: Depth 100 cm: Depth 110 cm: Depth 120 cm: Depth 130 cm: Depth 140 cm: Depth 15 cm: Depth 160 cm: Depth 17 cm: Depth 180 cm: Depth 19 cm: Depth 200 cm: Depth 210 cm: Depth 220 cm: Depth 230 cm: Depth 240 cm: Depth 250 cm: Depth 260 cm: Depth 270 cm: Depth 280 cm: Depth 290 cm: Depth 300 cm: Depth 310 cm: Depth 320 cm: Depth 330 cm: Depth 340 cm: Depth 350 cm: Depth 360 cm: Depth 37 cm: Depth 380 cm: Depth 390 cm: Depth 400 cm: Uolume 1.171328 cubic metres Volume 3.287629 cubic metres Uolume 5.992683 cubic metres Volume 9.153167 cubic metres Uolume 12.688680 cubic metres Volume 16.542607 cubic metres Uolume 20.671541 cubic metres Volume 25.040315 cubic metres Uolume 29.619312 cubic metres Volume 34.382846 cubic metres Volume 39.308129 cubic metres Volume 44.374558 cubic metres Uolume 49.563214 cubic metres Uolume 54.856491 cubic metres Volume 60.237812 cubic metres Uolume 65.691403 cubic metres Volume 71.202114 cubic metres Uolume 76.755259 cubic metres Volume 82.336490 cubic metres Uolume 87.931677 cubic metres Volume 93.526792 cubic metres Uolume 99-107811 cubic metres Uolume 104.660600 cubic metres Uolume 110.170809 cubic metres Volume 115.623754 cubic metres Uolume 121.004279 cubic metres Volume 126.296607 cubic metres Uolume 131.484157 cubic metres Volume 136.549317 cubic metres Uolume 141.473161 cubic metres Volume 146.235075 cubic metres Uolume 150.812257 cubic metres Uolume 155.179001 cubic metres Uolume 159.305663 cubic metres Volume 163.157031 cubic metres Uolume 166.689620 cubic metres Volume 169.846666 cubic metres Uolume 172.547399 cubic metres Volume 174.657114 cubic metres Uolume 175.743037 cubic metres

Program code to copy:

#define _CRT_SECURE_NO_WARNINGS

#include <stdio.h>
#include <math.h>
#include <stdlib.h>

// define the global variable
double width;
double height;
double length;

// declare a function function() to compute the
// cross section area of the tank
double function(double x);

// declare the function num_Integration_trapezoidal()
// to compute the intergration value by using trapezoidal method
double num_Integration_trapezoidal(double A, double B, int n);

// define the main function
int main()
{
   // define a variable to keep maintining the calibrated
   // units of the stick
   int calibratedUnit = 0;

   // define a variable to hold the upper bound of the
   // integration value
   int upperBound = 100;

   // declare the required variables
   double stickHeight, area_of_tank, volume_of_tank;

   printf("To compute the volume at the given depth at"
       " certain cross sectional area\n\n");
   // prompt the user for the width, height and length of the
   // tank in meters
   printf("Enter the data measurements in meters\n");

   // prompt and read the width of the tank
   printf("Enter tank width(m): ");
   scanf("%lf", &width);

   // prompt and read the height of the tank
   printf("Enter tank height(m): ");
   scanf("%lf", &height);

   // prompt and read the length of the tank
   printf("Enter tank length(m): ");
   scanf("%lf", &length);
   

   // at every calibratedUnit, find the area and volume and display the
   // volume cubic meters
   for (calibratedUnit = 10; calibratedUnit <= height * 100; calibratedUnit += 10)
   {
       // find the stick height
       stickHeight = calibratedUnit / 100.0;

       // call the num_Integration_trapezoidal() function
       // to find the area
       area_of_tank = num_Integration_trapezoidal(height/2.0-stickHeight,height/2.0,
           upperBound);

       // compute the volume of the tank
       volume_of_tank = area_of_tank * length;

       // display the depth and volume of the tank
       printf("Depth %d cm: Volume %.6lf cubic metres\n",
           calibratedUnit, volume_of_tank);
   }
   getchar(); // to halt the program
   getchar(); // to halt the program
   return 0;
}

// declare a function function() that accepts a double value
// and returns a double.
// This function is to compute the cross-section area of the tank.
double function(double x)
{
   double A = height / 2.0;
   double B = width / 2.0;
   double y = (B / A) * sqrt(A * A - x * x);
   return y;
}

// define a function num_Integration_trapezoidal() that accepts
// two double values and an integer value and returns a double value.
// This function is to find the intergrated value of the cross-section area
// by using trapezoidal method
double num_Integration_trapezoidal(double A, double B, int n)
{
   // declare the required variables
   double xVal, dx, totalSum = 0.0;
   double result = 0;
   int i = 0;

   // find the interval value of the integration
   dx = (B - A) / n;

   // find the initial sum value
   totalSum = (function(A) + function(B))/2;

   // add the value A with interval
   xVal = A + dx;

   // loop through each value of the interval till it reaches the
   // upper bound
   for (i = 1; i < n; i++)
   {
       // find the sum of the cross-section area
       totalSum += function(xVal);

       // increment the interval value
       xVal += dx;
   }

   // find the total area
   result = 2 * totalSum * dx;

   // return the result
   return result;
}