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Making a Can A can in the shape of a right circular

Algebra and Trigonometry | 9th Edition | ISBN: 9780321716569 | Authors: Michael Sullivan ISBN: 9780321716569 181

Solution for problem 87 Chapter 5

Algebra and Trigonometry | 9th Edition

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Algebra and Trigonometry | 9th Edition | ISBN: 9780321716569 | Authors: Michael Sullivan

Algebra and Trigonometry | 9th Edition

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Problem 87

Making a Can A can in the shape of a right circular cylinder is required to have a volume of 250 cubic centimeters. (a) Express the amount A of material to make the can as a function of the radius r of the cylinder. (b) How much material is required if the can is of radius 3 centimeters? (c) How much material is required if the can is of radius 5 centimeters? (d) Graph . For what value of r is A smallest?

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Week 1 Lecture GDP PER CAPITA HIGHER IN 20 TH CENTURY FOR ALL COUNTRIES EXCEPT: GERMANY – demanding patenting system, led to less patents granted RUSSIA – constant investment / GDP ratio fell over time (Rate of K stock hi) UK - all leaders in the previous century 4 Main sources of Growth: 1) Capital investment and accumulation 2) Gains from trade and specialization (“Smithian Growth”) 3) Improved allocation of the factors of production 4) Increased technological progress Before industrial growth: Most growth about 1-3 During industrial growth: solely about technological progress that is bolstered by the diffusion of knowledge – process and product innovation TFP: efficiency gains from some unknown factor (what is left after we deduct the inputs from the growth of output) Prime movers: energy converters that determined the movement and growth of society th Tech advances before the 19 Century were not due to scientific advances in knowledge 1870 – thermodynamics becomes a “mature science” 1869 – chemistry Kondratiev waves – PRDE SMIL REJECTS THE KONDRATIEV CYCLE + Age of synergy was unique + Not just like any cycle, so ignores dating advances by periodizations + 5 CHARACTERISTICS OF UNPRECEDENTED SALTATION (1. impact is instantaneous, 3. rate of improvement is so quick – zipper in 1851 and improvements in 1893 and 1912, 2. interconnectedness of scientific and technical advancements – electricity and steam --> electric motor and internal combustion engine, electromagnetic waves --> TV, telegraph, morse code, 4. Bold proposals, 5. ‘epoch- making’ era Greatest innovation between 1870-1914 TFP explains between 50-75% of per capita output growth Technological innovation accelerates labor productivity growth This analysis suggests the importance of expected profitability in justifying the high fixed costs of making investments / innovation in technology and perfecting good ideas to make them commercially viable è innovation in coke smelting, textiles and steam power require a lot of investment but rate of return on these innovations is much higher in Britain than anywhere else, so justifies the cost of the innovation è Conducted emplirical analysis on Britain vs. France’s profitability adopting new inventions in coke smelting, Hargreaves’ spinning jenny, Arkwright’s mill – in all cases, adoption more rational in Britain than France è Eventually, micro­inventions became a cumulative process and adoption spread to many countries ­­> leading to the Industrial Revolution Week 1 Reading Crafts & O’Rourke: Twentieth Century Growth *Reading focuses on: two key features that are important: (1) institutions (why they matter, how they work) and (2) the mechanics of technological change  Legacy of Industrial Revolution: led to interesting aftermath in the 19 Century – in terms of advent of economic growth and technological change The Beginnings of Modern Economic Growth + British economy particularly strong in industrial growth + Modern economic growth in mid­19 Century NOT characterized by increases in income  based on sustained labor productivity growth due to technological progress and higher levels of investment  Distinguish this from pre­industrial growth which was due to increases in working hours and Smithian growth + Modern economic growth ≠ increases in real income based on working harder and Smithian growth + TFP growth more than doubled from 0.3 to 0.7% from 1760­1801 to 1831­1873 + Education accounts for more than .3 percentage points of measured TFP growth from 1831­1873 + Technological innovation accelerates labor productivity growth LEARN Explanation for this: If contribution of technological change to labor productivity growth is considered to be capital deepening in the modernized sector plus TFP growth, then this equates to 0.54 out of 0.64% per year  this suggests that technological innovation accelerates labor productivity growth Many forms of technological advance can only take place when “embodied” in capital goods + Embodiment takes place in modernized sectors where we see technological changes embodied in capital goods  Paradox: despite strong tech advancements such as using steam as GPT (General Purpose Technology), TFP growth was modest – evidenced by stagnation of real wage rates (TFP growth = weighted average growth in factor rewards) Two takeaways from this: 1. Impact of technological progress was UNEVEN + Most of the service sector not impacted / unaffected except transport + In 1851, only 13.4% of industrial employment in textiles, metals and machine­making + Most industrial employment still in “traditional” sectors 2. Most technological improvements took a while to realize in an era where scientific understanding was limited EXAMPLE: STEAM  limited understanding; price fell because scientific potential of steam not understood fully – maximum impact of steam delayed until 3 rd th quarter of 19 Century After this, leading economies have become much better at exploiting GPTs due to: better scientific understanding, capital markets improvements, better education, government expenditure on R&D, thus greater rewards for innovative efforts. Directed Technical Change and the First Industrial Revolution Allen argues: the Industrial Revolution was invented in Britain because it paid to invent it there + Argument comes from an endogenous investment perspective + Based on relative factor prices and market size (rather than superiority of British institutions/policies) – based on demand for innovation and not the supply­side + High wages and cheap energy and sizeable market for new technologies ­­> conducive to innovation ^ This analysis suggests the importance of expected profitability in justifying the high fixed costs of making investments / innovation in technology and perfecting good ideas to make them commercially viable  innovation in coke smelting, textiles and steam power require a lot of investment but rate of return on these innovations is much higher in Britain than anywhere else, so justifies the cost of the innovation  Conducted emplirical analysis on Britain vs. France’s profitability adopting new inventions in coke smelting, Hargreaves’ spinning jenny, Arkwright’s mill – in all cases, adoption more rational in Britain than France  Eventually, micro­inventions became a cumulative process and adoption spread to many countries ­­> leading to the Industrial Revolution Contradictions / arguments against Allen’s hypothesis: 1) Important to consider the supply­side: stating that relative factor prices (Demand side) is only important in leading to the Industrial Revolution is bold 2) It may have been high cost of machinery rather than low wages in France that impeded the adoption of the spinning jenny in France (strikingly, would have been profitable to invent spinning jenny in high­wage US although market size is too small to justify high fixed development costs) High wages, cheap energy, and market size large enough to justify high fixed research & development costs ­­> conducive to accelerated technological progress (US more favoured location in 19 C ­­ leads in this after Britain, end of 19 Century, US overtakes) + Clear correlation between GDP and industrial output in 1913 + Crude TFP growth remained quite slow until Second Industrial Revolution (at th the end of 19 Century) Period before WWI Industry attracted to market potential Legislation that allows capital markets to function Most of Europe on its path to attaining open­access status Divergence Transition to modern economic growth happened first in Britain This led to a large increase in the gap between the industrial rich and non­industrial poor Industrialization first took hold in W Europe and then E Europe By 1913, Western European region income was 129% the world average – incomes rise rapidly due to industrialization The Great Specialization Falling transport costs and liberal trade policy – allowed for international division of labor – manufacturing costs decrease (rapidly during the Industrial Revolution) Great Specialization due to dramatically falling transport costs and geographically imbalanced technological progress ECON 324: Western Economic History Tuesday, 5 January 16 Class 1 + Countries were of different sizes then: territories in Europe + Europe after WWI + The layout and size of the countries have changed over time + Almost everything that could go wrong in Europe went wrong – yet made very little difference in terms of LT growth and rise of post- industrial society + Some evidence of convergence: started on very different GDPs and slowly, countries began to converge – this is an interesting debate in Economics. Is the world experiencing convergence – are we all moving towards the same standard of living + 1913 is a more integrated world than today only because for many countries (including US) there were no immigration restrictions – anyone is free to go to the country + The net rate of fertility is below replacement value – replacement value means that the rate of fertility is slightly over 2; the total number of people are in the population can be expected to give birth to if … Today, the net fertility rate in most European countries is very low (2.25). Average is ~1.1. + Fewer people being born and more people living longer, the median age is going up. Today, it is somewhere in the mid-40s. + Particularly after WWII, it goes through a particularly bad funk + Contracts can be enforced; good supply of infrastructure and water existed; society was in respectable order; freedom of movement and operations existed Tuesday, 12 Jan. 16 Class 3 Main areas of progress in the 2 ndIndustrial Revolution: Technology 1. Materials Steel: 2.5% carbon mixed in which maxmizes the resilience and tenacity functions of steel 2. Energy Electricity is prominent but many other developments 3. Chemistry – organic chemistry allows for applications such as dyes, fertilizers, ammonia 4. Mass production – this is the age in which modularity was not invented, but perfected – you make huge batches of guns or products together instead of separately – increases efficiency – makes a big difference for the military 5. Medicine: biggest discontinuity of all; what doctors do is limited by the fact that they are clueless about what makes people sick. People start realizing that when you’re sick it’s not because of bad air, lack of prayers, or phases of the moon – they realize that there are microorganisms in the air that enter your sytem. Antibiotics discovered many decades later but this is one of the most important discoveries in this age. 6. “Death of distance” techniques – invention of the telegraph Demographics + Variance declining over time in Sweden; peaks in mortality but they get smaller and smaller over time. 1925-1935 = Spanish flu. The big peaks/humps disappear because famines and big epidemics disappear. Baby boom was after too (1960-70ish) Quite a bit of variation within Europe 1. British Pattern 2. German Pattern – didn’t have a British hump but what is interesting is how late infant mortality rates start falling 3. French Pattern – fertility starts to decline before everyone does, no one actually knows why But in the end, there is a convergence to zero population or negative population growth Impact of the demographic transition on economic welfare You want to live in a society which has a low level instead of a high level of … population equilibrium + Pain of a losing a child is constant across all generations + One concern today parents don’t have is of children dying – indirtht effects on Human Capital – research + 20 century is the first time in history that we see peoplthenjoying Florida, retirement: The Golden Years. Nobody in the 19 Century would look forward to such a time. Fertility + Sharp decline in birth rates + Birth rates declining everywhere across all countries + If you calculate the coefficient of variation in 2000, it’s very small – birth rates are very similar in 2000 as compared to before (1890) + 1995 – total fertility rate – every single country is miles below replacement fertility. It almost looks like Europe is committing demographic suicide – they re not replacing themselves + Today, there is a little bit of bouncing back. The only country that is moving around replacement ironically is France – France had an extremely liberal set of policies around maternity leave, child subsidies, etc. in order to encourage higher rates. Ukraine’s numbers are truly scary + Two good reasons why we should have expected the reverse: 1. Naïve Darwinism: every species on this planet is subject to superfecundity. If there are no resource constraints, every species will grow vastly rapidly. What’s keeping numbers in check are resource constraints – you don’t have enough to eat or other species feast on you. Now, we are rich, we can eat as much as we want, and many resource constraints lifted – we do the reverse instead of what rabbits do. No one saw this coming because they expected humans to keep increasing with income. 2. Natural selection: shows a bunch of people don’t like kids. Two queries Children may be an inferior good, or prices went up (of having children), or something else changed. Possible solutions: there’s a few ways in which these queries have obvious solutions. We aren’t genetic programmed to like children, we are genetically programmed to like sex. Or we are programmed to be k-strategists (in biology, they distinguish between two types of species – k and r-strategists. R-strategists are ways in which the species have LOTS of offspring but don’t make it to adulthood like fish; k-strategists like elephants have one cub or one offspring and spend their entire lives raising it.) What mattered in the past was marital fertility – can be decomposed into various reasons: 1. Proportion ever conceiving 2. Age at first conception 3. Spacing 4. Age at “stopping” One major explanation of the fertility transition: the contraceptive revolution 1. Fertility control is based on two dimensions: ability and willingness to control fertility. 2. Read slides One simple but powerful explanation: reduced replacement conceptions, breastfeeding makes it less likely that she is going to be pregnant later (reduced fecundity). Fecundity – the probability that a woman becomes pregnant assuming constant sexual relations and no contraception so it is the “typical rate” on the graph. Famine and diseases are good reasons for fecundity, but this was not an issue – if fecundity is true, fertility should have risen. Thursday, 14 January 16 Paradoxes of the demographic transition + What is the price of having a child Unlike all other consumer goods, the full “price” of a child is not set parametrically by the market; much of it is determined by the parents, obviously constrained by market prices + Children as inferior goods – as women have become more educated and skilled; economies of scale (sibling effect) – there are major economies of scale in rearing children – if you look at an average home and all the children slept in one big bed, you add another child and toss them in with the siblings and they keep each other warm; people wear hand-me-downs – these days, a lot of siblings don’t wear hand-me-downs which is a loss of economies of scale. The younger children are being cared for by their older siblings so the marginal cost of the younger child is lower than the cost of the older child. Many of those economies of scale have disappeared in modern age (refuse to wear hand-me-downs, refuse to share a room with sibling, older siblings refuse to take care of younger ones) – what we see today is that much of the decline in fertility is because very few families have 10-12 children which was much more prevalent in the 19 century. Demand for children Children are a decision variable – if you did not want to have a large family, you would marry later (in the West), whereas in the East, they would just let the child die (infanticide). So, we can write this formally: K (number of children the family had) = K* (1 + mu) + e K* = number of children you want to be there when you’re old Mu = number of children you expect to die Epsilon = error term a. contraceptive failure (error term > 0) b. sterility (or a lack of fecundity) leading to error (error term > 0) c. spouse morbidity, mortality, and other forms of “separation” NO good markets for excess demand for children. What does this mean WHY Children as an investment good + Is the PV of a child > 0 + But leaves out: Bs and Cs are unknown in advance, Bs and Cs may differ for the two parents. Not only are the B’c and C’s unknown but they differ between the man and woman. If both parents disagree, they may bargain. Costs for woman > man. Women clearly have more bargaining power today than they did 100 years ago, so this could have brought the rate of fertility down. Coverture = legal doctrine stating that the married woman is not a legal entity. Married woman cannot own property; today, we live in a more enlightened age where the woman has more bargaining power, they have occupations, they have more bargaining power. Direct utility from children What exactly is it that parents get from having children + Companionship and need for continuity – demand for pets have gone up too – in the 19 Century, cats were there to catch mice and dogs existed to chase off burglars. The pet industry has ballooned in every Western country where the birth rate fell – we treat pets like children. + Dynastic utility – they want to continue the family and see themselves as custodians of their family lineage and want to know their roots. We care about our family lines and if we care about the past then we care about the future. + “Signaling” to others – we are a part of society and children send a certain signal to others. Having 12 children was an indication of a man’s manliness. Most beloved by Economists – read slides In the past, children were treated like adults. These historians say that we didn’t treat children differently than adults. If we love them more, why do we have fewer The quality issue may well be the key to the fertility decline. If you care about children and love your children, you should care about their income. - By investing in the human capital of your children, you make sure they are happier and richer. Maybe this will create some kind of affection and loyalty (and thus, it is an investment for old age). - Having a successful child reflects well on you – good genes, parenting, etc. Q-q trade-off (quality-quantity) is most popular among economists, but problematic + We should treat children as a discrete consumable like a car + Up in a richer neighbourhood, someone has a BMW and Mercedes (2 cars) and in a lower income neighbourhood, someone has a 20- year old run-down Chevy + some crappy car. Both have 2 cars but the quality of the cars differ. + As income goes up, you may have more money but there are still 24 hours in the day. Why can’t you hire a nanny to take care of your children + Paradox: If the q-q trade-off is the main reason why people have less children, you’ll expect it to be less of a problem in Europe than the US because education is free in Europe from K-12. Whereas in the US, education is expensive. IF this were the reason, you would expect fertility rates in Europe to be higher because quality is cheaper. But the birth rate in the US is higher, not lower – this paradox difficult to swallow. At some point, parents reach the red budget constraint because they’ve run out of time. The solution is that you end up with less children. This assumes that money and time are completely unsubstitutable. By and large, parents want to retain control and control takes time, which is why you face the budget constraint. How has the changing status of women affected fertility + Since 1945, the entire Western world has experienced two things: women have increasingly joined the labour force (before 1940, very few married women worked – either very poor or extremely highly educated women were a part of the workforce) – today, almost 2/3 or ¾ of the female population is working in Europe. Is it that women decided to have fewer babies and they have more time  job Or because norms changed and they want a career, they have fewer children Causality: want fewer babies --> job OR want job --> fewer babies Overall, was the fertility decline a “good thing” It may not be because populations are ageing, negative population growth in some places and it creates actuarial issues and there is labour scarcity. ON THE OTHER HAND, if you think of society as the quality went up – people are educated, better lives, better forms of leisure  fewer people can be preferred to a larger population where there is a competition for resources and against time. The principle of revealed preference indicates that by and large this is an improvement. By that logic, each person has made a decision to not have kids – contraceptive indicates this revealed preference and choice. The Great Mortality decline (shift from fertility to death now) th 1. Did not start in the 20 Century but much of it achieved after 1900. 2. Most prominent event: the decline of infectious diseases in Europe and the concomitant rise in life expectancy at birth from approx 35 years in 1800 to approx 76 years in 2000 (and approx 80 years in 2012). Nosology = the study of the cause of death. In the end it becomes an issue of knowledge – what do we know about diseases and how do we cure them The fight against death and disease The rule that governs medicine is known as Josh Billing’s Law. Mortality Decline table: e0 is life expectancy at birth Each disease has its own story Yet diseases also interacted with each other, through human immune system Yet all are susceptible to human interference through preventative and clinical channels. How much depends on medical science and technology, that is, human knowledge. Eastern Europe was typically worse than Western Europe; a long way to go in Haiti. In Afghanistan it’s 121 which is what Europe used to be 100 years ago. Their decline is steadier over time than other regions; even the 19 th Century in a industrialized country – it takes death rates a very long time to start declining, Cities until very recently are a bad place to live because the war is bad and the air is bad. Mortality decline model: Decline in mortality rates can have four types of causes 1. Increase in income and consumption 2. Changes in relative prices 3. Increases in private knowledge and health 4. Changes in public policy Health-improving good: ie. grapefruit juice (the more you drink the better) Health-indifferent good: income goes up, you’re richer, and as we’re richer we want to consume more of both goods and we consume more grapefruit juice because we enjoy it and the side effects are that we get healthier. McKweon Effect says we’re living longer because we’re richer. This can be tricky – 1. It is not guaranteed that higher income invariably implies better health. The bread and potatoes paradox implies something like this – if you’re really poor, you eat potatoes. If you were poor but not poor enough to eat potatoes, you ate coarse bread or barley bread. If you’re really rich, you would eat white/wheat bread – problem is that this is the least healthy for you. It turns out, as your income went up, you switched into less and less healthy foods – this defies McKweon’s theory. Relative price effect Suppose the health-improving good became cheaper than health- indifferent good – healthy foods become cheaper and also because there are some things where public work makes things cheaper (technological change in the public industry – water chlorination – seems obvious to us The most interesting phenomenon is improved consumer information – health itself is produced in the system by which goods are consumed. We can improve our consumption bundle by which goods are consumed. Europe suffers from malaria – until 1900, no one knews that mosquitoes transmitted malaria (mal aria = bad air). They didn’t know and people figured it out and once it became known, the consumer information effect came into play. Some diseases are caused by nutritional deficiencies but nobody before 1900s had a clue. The absence of vitamin C (scurvy) doesn’t exist now – in the 19 Century, scurvy was common. Information greatly improves health in society. IF you’re poor enough and you live a lousy diet, you can suffer from deficiencies but then people figured out – at a really low cost, you can acquire enough vitamin C to rule out scurvy forever. Cabbage worked just as well (for poor people) – so the disease was eliminated. It will clearly do this kind of thing (consumer information effect). Public Health A clearly important factor affecting mortality. Read PPT slide Public health improved sanitary policy and led to fewer health issues. A whole bunch of things that the public sector is more effective in solving than individuals are made clear. Private Health On one hand, we have the municipal government doing things but people changed their consumption packets so people bought more detergents, insecticide, clean water, cod liver oil, and other means of preventing diseases. Public and private health work hand-in-hand. The questions that eluded people for 10,000 years were slowly being answered and solved in our history. Three Sources of Medical Knowledge (how are people learning to improve health) 1. Statistical method and hygiene – looking for empirical regularities. 2. Germ theory: people discover how diseases are being transmitted. 3. Nutritional theories of health The Germ Theory created an interesting problem + Better cleaning + Disinfecting things + Washing things + People changed their consumption and behavior patterns – boiling things A lot of the actual impact depended on dissemination and persuasion. You have to persuade them to change their behavior – so hard to get the world to stop smoking. Between 1880-1914, there was a major effort to disseminate and diffuse “sanitary knowledge”. Doctors made house calls and the doctor became the household consultant to prevent the onset of further disease. By 1930 the process was complete. And the next thing we saw was a miraculous thing. Next week, WWI.

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Chapter 5, Problem 87 is Solved
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Textbook: Algebra and Trigonometry
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716569

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Making a Can A can in the shape of a right circular