Applications of ode and matrices
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The document covers mathematical concepts including exponential growth and decay, radioactive decay, and the falling object problem. It explains the formulas related to bacterial culture growth, half-life calculations for radioactive elements, and the physics of falling objects under gravity. Additionally, it discusses cryptography processes for encryption and decryption along with geometric concepts like the area of a triangle and collinearity of points.
![The easiest scheme is to let space=0, A=1, B=2,Y=25, and Z=26.
• Encrypt the word REDRUM by using matrix [B] as encryption.
B =
• The message "Red Rum" would become 18, 5, 4, 0, 18, 21, 13.
4 −2
−1 3](https://image.slidesharecdn.com/applicationsofodeandmatrices-140611091350-phpapp01/85/Applications-of-ode-and-matrices-14-320.jpg)
Applications of ode and matrices
- 2. Faizan Shabbir 13-ME-032 Shameel Shahid Hashmi 13-ME-158 Ibrahim Munir 13-ME-102 Muhammad Ali Khan 13-ME-071 Danial Zafar Gondal 13-ME-027 Khurram Shahzad 13-ME-059
- 3. Exponential Growth & Decay Radioactive Decay Falling Object Problem
- 4. • Exponential functions come into in situations in which the rate at which some quantity grows or decays. • As we are taking about rate of change of some quantity. Therefore, 𝒅𝒚 𝒅𝒕 ∝ y And final form is y = y°ekt
- 5. It is known that cells of a given bacterial culture divide every 3.5 hours (on average). If there are 500 cells in a dish to begin with. How many will there be after 12 hours?
- 6. • The half life of a radioactive element is the time required for half of the radioactive nuclei present in the sample to decay. • For the quantity to reduce one half • According to differential equation 𝒅𝒚 𝒅𝒕 = 𝒌𝒚 Let y0 be the number of radioactive nuclei present initially.Then the number y of nuclei present at time t will be given by y = y°ekt 𝒉𝒂𝒍𝒇 𝒍𝒊𝒇𝒆 = 𝒍𝒏 𝟐 −𝒌
- 7. 1) The number of atoms of plutonium-210 remaining after t days, with an initial amount of y0 radioactive atoms, is given by: y = y°e(-4.95x10-3)t Find the half-life of plutonium-210. 2) Uranium 237 has a half-life of about 6.78 days. If there are 10 grams of Uranium 237 now, how much will be left after 2 weeks?
- 8. If a stationary object at height y0 is dropped and is not pushed with any force. Gravity causes the object to fall and to accelerate as it falls. Air resistance will be ignored. We use the fact that the acceleration due to the downward force of gravity is g = −9.81m/s2 = −32ft/s2. (Note:The acceleration is considered to be negative because the motion of the object is downward.)
- 9. We use the following facts: • Acceleration is the rate of change in velocity over time so, 𝒅𝒗 𝒅𝒕 = 𝒂 Since, with no air resistance, the acceleration of the object is constant at g, we get the mathematical model 𝑑𝑣 𝑑𝑡 = g • Velocity is the rate of change in position over time, so 𝑣 = 𝑑𝑦 𝑑𝑡 V (0) = 0 , y (0) = yo
- 10. An object is dropped from a height of 500 m.When will the object reach ground level, and with what speed?
- 11. Cryptography Area of a Triangle Test for Collinear Points
- 12. It involves two processes: • Encryption Process • Decryption Process
- 13. • Convert the text of the message into a stream of numerical values. • Place the data into a matrix. • Multiply the data by the encoding matrix. • Convert the matrix into a stream of numerical values that contains the encrypted message.
- 14. The easiest scheme is to let space=0, A=1, B=2,Y=25, and Z=26. • Encrypt the word REDRUM by using matrix [B] as encryption. B = • The message "Red Rum" would become 18, 5, 4, 0, 18, 21, 13. 4 −2 −1 3
- 15. • Place the encrypted stream of numbers that represents an encrypted message into a matrix. • Multiply by the decoding matrix.The decoding matrix is the inverse of the encoding matrix. • Convert the matrix into a stream of numbers. • Convert the numbers into the text of the original message.
- 16. Perform decryption of following numbers. 67, -21, 16, -8, 51, 27, 52, -26. Encryption matrix is B = 4 −2 −1 3
- 17. • Find vertices (x1,y1), (x2,y2), and (x3,y3) of triangle in (X,Y) plane • Use the formula Area = ± 1/2 • No matter answer comes positive or negative. Always take positive value for area. X1 Y1 1 X2 Y2 1 X3 Y3 1
- 18. • Find area of the following triangle.
- 19. • Find X &Y coordinates of points • If the equation satisfies. It means points are collinear. x1 y1 1 x2 y2 1 = 0 x3 y3 1
- 20. Check if points A,B,C are collinear. • B = (9,48) • A = (3,18) • C = (6,33)