Real Life Applications of Mathematics.pptx

Applications

1. M.K. TAMIL SELVI HOD, MATHS Alpha College of Engineering

2. Mathematics is the gate and key of the sciences... Neglect of mathematics works injury to all knowledge , since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus Ignorant are unable to perceive their own ignorance and so do not seek a remedy. Roger Bacon

3. At Home Setting an alarm and hitting snooze, they may quickly need to calculate the new time they will arise. Or they might step on a bathroom scale and decide that they’ll skip those extra calories at lunch. People on medication need to understand different dosages, whether in grams or milliliters. Recipes call for ounces and cups and teaspoons --all measurements, all math. And decorators need to know that the dimensions of their furnishings and rugs will match the area of their rooms.

4. In Travel Travelers often consider their miles-per-gallon when fueling up for daily trips, but they might need to calculate anew when faced with obstructionist detours and consider the cost in miles, time and money. Air travelers need to know departure times and arrival schedules. They also need to know the weight of their luggage unless they want to risk some hefty baggage surcharges. Once on board, they might enjoy some common aviation-related math such as speed, altitude and flying time.

5. SOME COMMON VIEWS OF MATHEMATICS MATHS IS HARD MATHS IS BORING MATHS HAS NOTHING TO DO WITH REAL LIFE ALL MATHEMATICIANS ARE MAD! BUT I CAN SHOW YOU THAT MATHS IS IMPORTANT IN CRIME DETECTION, MEDICINE and ......

6. Balancing the checkbook. Understanding loans for various purposes. Understanding sports Managing money. Shopping for the best price. Figuring out time, distance, and cost for travel. Playing music. Gardening and landscaping. There are several concepts that include maths, such as weighing, understanding chemical formulas, analyzing marketing data, measuring, drafting, and calculating statistics.

7. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009? It can be solved quickly using differential equations BIRTH

8. Temperature of a dead body at 11:30 pm was 94.6 F. Temperature of the body again after one hour , which was 93.4 F. If the temperature of the room was 70 F, estimate the time of death. Taking normal temperature of body as 98.6 F. This also can be solved using differential equations. DEATH

9. MATHS AND CRIME A short mathematical story Burglar robs a bank Escapes in getaway car Pursued by police GOOD NEWS Police take a photo BAD NEWS Photo is blurred

10. Original Blurred

11. Original f(x) g(x) Blurring h(x) = f(x)*g(x) SOLUTION Maths gives a formula for blurring convolution By inverting the formula we can get rid of the blur

12. Modern medicine has been transformed by methods of seeing Inside you without cutting you open! MATHS AND MEDICINE Ultra sound: sound waves MRI: magnetism CAT scans: X rays

13. WHAT IS A CAT SCAN?? CAT = Computerised axial tomography Based on X-Rays discovered by Roengten X-Rays cast a shadow GOOD for looking at bones BAD for looking at soft tissue

14. USING MODERN MATHS WE CAN DO A LOT BETTER Modern CAT scanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

15. Intensity of X-ray at detector depends on width of object We can find the thickness … can we find the shape?

16. MOVE SOURCE AND DETECTOR AROUND GET SHADOWS OF THE OBJECT FROM MANY ANGLES AND MEASURE X-RAY INTENSITY

17. Measure attenuation of X-Ray R(ρ, θ)

18. The study of the art involves the study of geometry; therefore, the students who have knowledge of basic geometry formulas can easily craft impressive art features. Also, each photographer uses mathematics to measure the focal length, exposure time, shutter speed, and lighting angles to take the photos. ARTS

19. Sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. A graphical representation of music allows computers to create and understand sounds. It also allows sound engineers to visualize sound waves so that they can adjust volume, pitch and other elements to create the desired sound effects. Trigonometry plays an important role in speaker placement as well, since the angles of sound waves hitting the ears can influence the sound quality. MUSIC

20. CRIMINOLOGY – Blood Stain Analysis

21. To find out how light levels at different depths affect the ability of algae to photosynthesize. Trigonometry is used in finding the distance between celestial bodies. Utilize mathematical models to measure and understand sea animals and their behaviour. Marine biologists may use trigonometry to determine the size of wild animals from a distance. MARINE

22. APPLICATIONS IN ENGINEERING If you know the distance from where you observe the building and the angle of elevation you can easily find the height of the building. Similarly, if you have the value of one side and the angle of depression from the top of the building you can find and another side in the triangle, all you need to know is one side and angle of the triangle.

23. APPLICATIONS IN ENGINEERING Game, Mario. When you see him so smoothly glide over the road blocks. He doesn’t really jump straight along the Y axis, It is a slightly curved path or a parabolic path that he takes to tackle the obstacles on his way. Trigonometry helps Mario jump over these obstacles. Gaming industry is all about IT and computers and hence Trigonometry is of equal importance for these engineers.

24. APPLICATIONS IN ENGINEERING Flight engineers have to take in account their speed, distance, and direction along with the speed and direction of the wind. The wind plays an important role in how and when a plane will arrive where ever needed this is solved using vectors to create a triangle using trigonometry to solve. Trigonometry will help to solve for that third side of your triangle which will lead the plane in the right direction, the plane will actually travel with the force of wind added on to its course.

25. Consider problem of a substance dissolved in a liquid. The liquid entering the tank may or may not contain more of the substance dissolved in it. Liquid leaving the tank will contain the substance dissolved in it. If Q(t) gives the amount of the substance dissolved in the liquid in the tank at any time t  To develop a differential equation, when solved, will give us an expression for Q(t). MIXING PROBLEMS Assumption: Concentration of the substance in the liquid is uniform throughout the tank.

26. The main “equation” for this model

27. A 1500 gallon tank initially contains 600 gallons of water with 5 lbs of salt dissolved in it. Water enters the tank at a rate of 9 gal/hr and the water entering the tank has a salt concentration of 15(1+cos(t)) lbs/galI. If a well mixed solution leaves the tank at a rate of 6 gal/hr, how much salt is in the tank when it overflows? MIXING PROBLEMS

28. Assume a uniform concentration of salt in the tank. The concentration at any point in the tank and hence in the water exiting is given by, 𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛= (𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑠𝑎𝑙𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎𝑡 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒 𝑡,)/(𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎𝑡 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒, 𝑡) The amount of salt at any time t is Q(t). Let the initial volume be 600 gallons Every hour 9 gallons enter and 6 gallons leave. At any time t there is 600+3t gallons of water in the tank.

29. An apple pie with an initial temperature of 170 C is removed from the oven and left to cool in a room with an air temperature of 20 C. Given that the temperature of the pie initially decreases at a rate of 3:0 C=min, how long will it take for the pie to cool to a temperature of 30 C? Assuming the pie obeys Newton's law of cooling, we have the following information: 𝑑𝑇/𝑑𝑡=−𝑘(𝑇−20), T(0) = 170; T’(0) = -3:0; where T is the temperature of the pie in celsius, t is the time in minutes, and k is an unknown constant. ∫1𝑑𝑇/(𝑇−20)=−∫𝑘𝑑𝑡 ⟹ ln⁡|𝑇−20|=−𝑘𝑇+𝐶. 𝑎𝑛𝑑 𝑇=20+𝐶𝑒^(−𝑘𝑡) 〗 Solving, 𝑇=20+150𝑒^(−0,02𝑡) COOKING

30. The Math of Social Distancing Is a Lesson in Geometry Determining how to safely reopen buildings and public spaces under social distancing is in part an exercise in geometry: If each person must keep six feet away from everyone else, then figuring out how many people can sit in a classroom or a dining room is a question about packing non-overlapping circles into floor plans.

31. Fibonacci numbers occur in nature in many places. Petals of flowers are an example While individual examples may disagree, the most common number of petals are usually close to Fibonacci numbers Mathematics determined the symmetry, the harmony, the eye's pleasure. FIBONACCI NUMBERS

32. When you count the number of petals of flowers in your garden, you will get the numbers 3, 5, 8,13, 21, 34, or 55. These numbers are not random numbers. These are very unique numbers and all of them part of Fibonacci sequence, which are series of numbers developed by a 13th century mathematician. You can also get the same numbers if you start with the numbers 1 and 1. And from that point on you keep adding up the last two numbers. 1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5=8 and you keep going like this. You will get the number of the petals of flowers.

33. Geometric sequences have a domain of only natural numbers (1,2,3,...), and a graph of them would be only points and not a continuous curved line. The best one is tile values in the game 2048. tile 1 = 2 tile 2 = 4 tile 3 = 8 tile 4 = 16 tile 5 = 32 tile 6 = 64 And so on GAME 2048

34. Compute the sum of all tiles Easy…… That’s 4+8+16+…+131,072, right? As a lazy mathematician, I don’t like doing long additions… So let me show you a trick to easily compute the sum of this all. Let me add 4 to the sum  4+8+16+…+131,072 By combining 4 with our tile 4, we obtain a new tile 8. But then, this new tile 8 can be combined with the existing tile 8, hence creating a new tile 16… And so on. Amazingly, our computation simplifies to the simple addition 131,072+131,072. Now add 4 to 4+8+16+…+131,072 to obtain that number. So to determine  4+8+16+…+131,072 we merely need to subtract 4 to the computation we did above, hence obtaining 4+8+16+…+131,072=(131,072+131,072)-4=262,140 How awesome is that? Geometric Sums in 2048

35. Waw! That’s magical! But where does that come from? It’s the magic of so-called geometric sums. Our geometric sum is 2^2+2^3+2^4+…+2^17 and we have the crucial property 2^𝑘+2^𝑘=2^(𝑘+1) So, by adding 2^2, we have 2^2+ (2^2+ 2^3+2^4+…+2^17 ) =(2^2+2^2 )+( 2^3+2^4+…+2^17 ) = 2^3 +〖(2〗^3+2^4+…+2^17)

36. The Health Canada has been monitoring flu outbreaks continuously over the last 100 years. They have found that the number of infections follows an annual (seasonal) cycle and a twenty-year-cycle. In all, the number of infections I(t) are well-approximated by the function: FLU-VACCINATION

37. t is measured in months and I(t) given in units of 100, 000 individuals. The Figure depicts the number of infections I(t) over time and illustrates the superposition of the annual cycle of seasonal flu outbreaks modulated by slower fluctuations with a longer period of 10 years

38. The minimum of the 5-year average by solving (𝑑𝐼 ̅(𝑡))/𝑑𝑡=0

39. Note, we have used the trigonometric identity cos(α+β)=cos αcosβ-sinαsinβ Hence, the start of a 5-year minimum (or maximum) average period is marked by the condition 𝑐𝑜𝑠(𝜋/120 𝑡)=−𝑠𝑖𝑛(𝜋/120 𝑡). Now cos α = -sin α holds for 𝛼=3𝜋/4 and 𝛼=7𝜋/4 (as well as when adding multiples of 2π to α). Since cos(α) is a decreasing function for 0 < α < π, we expect that 𝛼=3𝜋/4 marks a minimum. 𝜋/120 𝑡=3𝜋/4. Yields t=90 months. Indeed, this indicates the start of a 5-year minimum average because ├ (𝑑^2 𝐼 ̅(𝑡))/〖𝑑𝑡〗^2 ┤|_(𝑡=90)=𝜋/120>0. Hence the earliest intervention could start on June 1st , 2020.

40. Mathematics puzzles, games, Government and military organization websites Financial information like credit card number and bank account, Information security, all related encode, decode, theory MATRICES

41. For cooking or baking anything, a series of steps are followed, telling us how much of the quantity to be used for cooking, the proportion of different ingredients, methods of cooking, the cookware to be used, and many more. Such are based on different mathematical concepts. Indulging children in the kitchen while cooking anything, is a fun way to explain maths as well as basic cooking methods. Maths in kitchen

42. MATHS in sports Maths improves the cognitive and decision-making skills of a person. Such skills are very important for a sportsperson because by this he can take the right decisions for his team. If a person lacks such abilities, he won’t be able to make correct estimations. So, maths also forms an important part of the sports field.

43. Management of Time Now managing time is one of the most difficult tasks which is faced by a lot of people. An individual wants to complete several assignments in a limited time. Not only the management, some people are not even able to read the timings on an analog clock. Such problems can be solved only by understanding the basic concepts of maths. Maths not only helps us to understand the management of time but also to value it. Logical Reasoning Basic Mathematical Operations Reasoning

44. Maths is the basis of any construction work. A lot of calculations, preparations of budgets, setting targets, estimating the cost, etc., are all done based on maths. If you don’t believe, ask any contractor or construction worker, and they will explain as to how important maths is for carrying out all the construction work. In Construction Preparing budgets Estimating the cost and profit Arithmetic calculations Geometry Calculus and Statistics Trigonometry

45. Interior Designing Interior designing seems to be a fun and interesting career but, do you know the exact reality? A lot of mathematical concepts, calculations, budgets, estimations, targets, etc., are to be followed to excel in this field. Interior designers plan the interiors based on area and volume calculations to calculate and estimate the proper layout of any room or building. Such concepts form an important part of maths. Geometry Ratios and Percentages Mathematical Operations Calculus and Statistics

46. ‘Speed, Time, and Distance’ all these three things are studied in mathematical subjects, which are the basics of driving irrespective of any mode of transportation. Logical reasoning Numerical Reasoning Mathematical Operations Driving

47. Math is good for the brain Math helps you with your finances Math makes you a better cook Better problem-solving skills Every career uses math Great career options Math for Fitness Helps you understand the world better Time management To Save Money TOP 10 IMPORTANCE OF MATHEMATICS IN EVERYDAY LIFE