COMP 208: Computers in Engineering
COMP 208: Computers in Engineering Fall 2012 Assignment 6 Depth Gauge Integrity This assignment is to be done individually. You can collaborate on understanding the problem but you must write the solution individually. Your submission might be subject to plagiarism detection software. Due Date Assignment 6 is due on Wednesday, December 5 at midnight. However assignments will be accepted without penalty until Saturday, December 8 at midnight. The cutoff is automated and is exactly at this time. 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. The assignment is divided into two parts. Part 1: (40 marks) 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 centimetres), 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. 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 (60 marks) 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 Volume Volume . Volume Volume Volume
5: Depth 26.54 cm 10: Depth 42.48 cm 15: Depth 56.08 cm . . 165: Depth 355.12 cm 170: Depth 370.52 cm 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 numerical integration. Write a program to determine the roots of this equation for each depth by using a root finding algorithm such as the bisection method. Each function evaluation will have to be done using an integration algorithm such as the trapezoidal method.
Requirements Your code must meet these requirements: The program must be written in C Your program can write to the standard output or to a file. Use sensible variable names. Comment and indent your code Submit only the two .c files. Don’t submit the .exe (name your source files A6P1_123456789.c and A6P2_123456789.c where 123456789 is your ID) If any of the above requirements is not respected you will lose marks.
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. The assignment is divided into two parts. Part 1: (40 marks) 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 centimetres), 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. 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 (60 marks) 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 Volume Volume . Volume Volume Volume
5: Depth 26.54 cm 10: Depth 42.48 cm 15: Depth 56.08 cm . . 165: Depth 355.12 cm 170: Depth 370.52 cm 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 numerical integration. Write a program to determine the roots of this equation for each depth by using a root finding algorithm such as the bisection method. Each function evaluation will have to be done using an integration algorithm such as the trapezoidal method.
Requirements Your code must meet these requirements: The program must be written in C Your program can write to the standard output or to a file. Use sensible variable names. Comment and indent your code Submit only the two .c files. Don’t submit the .exe (name your source files A6P1_123456789.c and A6P2_123456789.c where 123456789 is your ID) If any of the above requirements is not respected you will lose marks.
