Solving Problems With a Distance-Rate-Time Formula

In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. The rate is the speed at which an object or person travels. It is usually denoted by  r  in equations.  Time is the measured or measurable period during which an action, process, or condition exists or continues. In distance, rate, and time problems, time is measured as the fraction in which a particular distance is traveled. Time is usually denoted by t in equations.   Solving for Distance, Rate, or Time When you are solving problems for distance, rate, and time, you will find it helpful to use diagrams or charts to organize the information and help you solve the problem. You will also apply the formula that solves distance, rate, and time, which is  distance rate x time. It is abbreviated as: d rt There are many examples where you might use this formula in real life. For example, if you know the time and rate a person is traveling on a train, you can quickly calculate how far he traveled. And  if you know the time and distance a passenger traveled on a plane, you could quickly figure the distance she traveled simply by reconfiguring the formula. Distance, Rate, and Time Example Youll usually encounter a distance, rate, and time question as a  word problem  in mathematics. Once you read the problem, simply plug the numbers into the formula. For example, suppose a  train leaves Debs house and travels at 50 mph. Two hours later, another train leaves from Debs house on the track beside or parallel to the first train but it travels at 100 mph. How far away from Debs house will the faster train pass the other train? To solve the problem, remember that d represents the distance in miles from Debs house and t  represents the time that the slower train has been traveling. You may wish to draw a diagram to show what is happening. Organize the information you have in a chart format if you havent solved these types of problems before. Remember the formula: distance rate x time When identifying the parts of the word problem, distance is typically given in units of miles, meters, kilometers, or inches. Time is in units of seconds, minutes, hours, or years. Rate is distance per time, so its units could be mph, meters per second, or inches per year. Now you can solve the system of equations: 50t 100(t - 2) (Multiply both values inside the parentheses by 100.)50t 100t - 200200 50t (Divide 200 by 50 to solve for t.)t 4 Substitute t 4 into train No. 1 d 50t 50(4) 200 Now you can write your statement. The faster train will pass the slower train 200 miles from Debs house. Sample Problems Try solving similar problems. Remember to use the formula that supports what youre looking for—distance, rate, or time. d rt (multiply)r d/t (divide)t d/r (divide) Practice Question 1 A train left Chicago and traveled toward Dallas. Five hours later another train left for Dallas traveling at 40 mph with a goal of catching up with the first train bound for Dallas. The second train finally caught up with the first train after traveling for three hours. How fast was the train that left first going? Remember to use a diagram to arrange your information. Then write two equations to solve your problem. Start with the second train, since you know the time and rate it traveled: Second traint x r d3 x 40 120 milesFirst traint x r d8 hours x r 120 milesDivide each side by 8 hours to solve for r.8 hours/8 hours x r 120 miles/8 hoursr 15 mph Practice Question 2 One train left the station and traveled toward its destination at 65 mph. Later, another train left the station traveling in the opposite direction of the first train at 75 mph. After the first train had traveled for 14 hours, it was 1,960 miles apart from the second train. How long did the second train travel? First, consider what you know: First trainr 65 mph, t 14 hours, d 65 x 14 milesSecond trainr 75 mph, t x hours, d 75x miles Then use the  d rt  formula as follows: d (of train 1) d (of train 2) 1,960 miles75x 910 1,96075x 1,050x 14 hours (the time the second train traveled)

Macbeth and Lady Macbeth as A Dead Butcher and His...

Macbeth and Lady Macbeth as A Dead Butcher and His Fiend-like Queen in William Shakespeares Macbeth Macbeth is one of Shakespeares four famous tragedies. It was written in 1605-1606, at the peak of Shakespeares career; and was chosen to accolade the new King James I of England, who had been James VI of Scotland. He had a fascination with witchcraft and the supernatural, so the play Macbeth complimented his passion. Shakespeare is famed for his use of the English language, turning nouns to verbs and verbs to nouns, and adding new words and expressions to the English language. His play, Macbeth is most famous for its brutal insights into characters and the impact of good and evil. At the†¦show more content†¦This is showing us that before he meets the witches, or is imposed upon by Lady Macbeth, Macbeth is a valiant man, who is loyal to his King and country. Macbeths first reaction to the witchs prophecies is to kill the King so that he shall be crowned. However although this appears to be a butcher- like plan, Macbeth soon has doubts on the situation. His admittance that Duncan hath honoured me of late, and that Hes here in double trust, show the guilt that Macbeth would feel and the loyalty he still possesses towards his king. He tells Lady Macbeth that we will proceed no further in this business . However he is too easily persuaded by his wife to continue in their evil plan. Macbeths selfish nature begins to be shown here. He only agrees to commit Duncans murder when another shall bear the guilt. This seems the easy option for Macbeth as he can obtain his title as king and shift the blame and consequences to somebody else, he is happy for others to suffer for his own aid. Our first impression of Lady Macbeth is slightly different. Our first encounter with her is when she reads a letter from Macbeth, describing to her his experience with the three witches and his recent events. 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Therefore the Read MorePlay Macbeth11985 Words   |  48 PagesAt the beginning of the play, Macbeth is a respected general, a devoted husband, and a loyal subject of the king. The first of the witches prophecies bring out his ambitious nature, but he struggles with killing the king. By attacking his manhood, Lady Macbeth convinces him to committ the first of his evil deeds. Macbeths evil deed causes him to suffer from fear and guilt, which leads to even more evil crimes. Then Macbeth becomes paranoid, suffering from hallucinations and sleeplessness. He becomesRead MorePlay Macbeth11979 Words   |  48 PagesAt the beginning of the play, Macbeth is a respected general, a devoted husband, and a loyal subject of the king. The first of the witches prophecies bring out his ambitious nature, but he struggles with killing the king. By attacking his manhood, Lady Macbeth convinces him to committ the first of his evil deeds. 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Applications of Nanocellulose Free Essays

an overview of the recent research on the fundamental and applied properties of nanoparticles extracted from cellulose, the most abundant polymer on the planet and an essential renewable resource. Given the rapid advancements in the field and the high level of interest within the scientific and industrial communities, pioneered the use of cellulose nanoparticles (cellulose nanocrystals or whiskers and cellulose microfibrils) in nanocomposite applications. n the life sciences and bio-based applications, biological, chemical and agricultural engineering, organic chemistry and materials science. We will write a custom essay sample on Applications of Nanocellulose or any similar topic only for you Order Now Cellulose has great potential as a nanomaterial as it’s abundant, renewable and biodegradable. It can be used in paper for its superior strength properties and can also be used as a wet-end additive to enhance retention in coating and packaging applications. Nanocellulose can form transparent films with excellent barrier properties, allowing it to be competitive with petroleum-based plastics in food packaging. Due to its reinforcing properties, nanocellulose can also be used in bio-composites and other matrix materials. 6. 9 Concluding remarks Natural fibres, cellulose and other constituents of natural fibres are very promising materials for the future, having the capability to replace current synthetic materials. ith the rapid developments in nanotechnology, nanocellulose brings many new insights to the materials world, such as its modulus value of 160 GPa, which is much greater than metallic materials. Considerim the challenges mentioned earlier, it is rational to predict that immediate applications of nanocellulosic materials can be formulated from water-based polymer matrices, like polyvinyl alcohol, starch/polyvinyl alcohol blends and latexes, by which two major problems can be eliminated. Here, the dispersion is done in the aqueous phase, so the additional step of drying can be omitted and the hydrophobic modification of the nanocellulose materials is not required. However, the main limitation is that this will give a product that can be used only tinder dry condition like This paper provides an overview of recent progress made in the area of cellulose nanofibre-based nanocomposites. An introduction into the methods used to isolate cellulose nanofibres (nanowhiskers, nanofibrils) is given, with details of their structure. Following this, the article is split into sections dealing with processing and characterisation of cellulose nanocomposites and new developments in the area, with particular emphasis on applications. The types of cellulose nanofibres covered are those extracted from plants by acid hydrolysis (nanowhiskers), mechanical treatment and those that occur naturally (tunicate nanowhiskers) or under culturing conditions (bacterial cellulose nanofibrils). Research highlighted in the article are the use of cellulose nanowhiskers for shape memory nanocomposites, analysis of the interfacial properties of cellulose nanowhisker and nanofibril-based composites using Raman spectroscopy, switchable interfaces that mimic sea cucumbers, polymerisation from the surface of cellulose nanowhiskers by atom transfer radical polymerisation and ring opening polymerisation, and methods to analyse the dispersion of nanowhiskers. The applications and new advances covered in this review are the use of cellulose nanofibres to reinforce adhesives, to make optically transparent paper for electronic displays, to create DNA-hybrid materials, to generate hierarchical composites and for use in foams, aerogels and starch nanocomposites and the use of all-cellulose nanocomposites for enhanced coupling between matrix and fibre. A comprehensive coverage of the literature is given and some suggestions on where the field is likely to advance in the future are discussed. How to cite Applications of Nanocellulose, Papers