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Differential Calculus: Maximum and Minimum Problem and Solution

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $200,000 per km overland to a point P on the north bank and $400,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places). 1 year ago Report Abuse Colorado… Best Answer – Chosen by Voters.
O will be the x-axis origin, P is the point on the north bank, and x= distance from O to the storage tanks. [Note, we could have put R at the origin, but the algebra is a little simpler this way]. The cost C(x) of the pipeline as a function of x is:
C(x) = distance along north shore * pipeline cost overland + distance under the river * pipeline cost under land The distance along the north shore is 6-x.

The distance (by Pythagorean theorem) under the water is sqrt:
( 2^2 + x^2) So, C(x) = (6-x)*200000 + sqrt(4 + x^2) * 400000
[You should graph this]. To find the value of x where C(x) is minimized, we set dC/dx = 0, [Reminder – use the chain rule to differentiate the second term]. Differentiating and simplifying, we getdC/dx = C'(x) = -200000 + 400000x/ sqrt(4+x^2) = 0 400000x / sqrt(4+x^2) = 200000 400000x/200000 = sqrt(4+x^2). Squaring both sides, we get 4x^2= 4 + x^2 x = sqrt(4/3) = 1. 15 So the distance from the refinery to point P is 6-x = 4. 85 km.

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The Problem of Thor Bridge

Other conventions are used in the text yet the previously sites will be discussed in this essay. All of these conventions are evident through Dole’s literary features. These crime writing features ensure the text directly conforms to the great detective style of the genre. The text begins with, Watson the central character, detective Sherlock Holmes loyal assistant narrating the previous case they had tackled. This convention of Watson narration is demonstrated in the first line “there is a travel-worn and battered tin dispatch-box with my name, John H.
Watson, M. D”. He then goes on to tell of the achieving of a letter from a clearly distressed male asking for help to protect his girlfriend, reflecting on the convention of a man in distress begging for help with a case. “l can’t explain things-I can’t even try to explain them, but I know beyond all doubt that Miss Dunbar is innocent. ” The literary technique, first person perspective is used throughout the letter to describe the situation so that it becomes apparent to the reader that a real man is pleading for help from the talented detective.
The story continues as Holmes begins to use his incredible skills of observation to covers clues, when he notices a mark on Thro Bridge, this alone demonstrates the distinctive characteristics of the great detective. The description in the quote “The stonework was grey, but at this one point It showed white for a space not larger than a sixpence,” it draws attention to his superior surveillance talent. Nearing the conclusion of the story Holmes communicates his discoveries to those partaking In the Incident.

This convention Is demonstrated In the final paragraph of the short story “Then she took one of her husband’s revolvers-there was, as you saw, n arsenal In the house-and kept It for her own use. A similar one she concealed that morning In Miss Dunbar wardrobe after discharging one barrel, which she could easily do In the woods without attracting attention. She then went down to the bridge where she had contrived this exceedingly Ingenious method for getting rid of her weapon. When Miss Dunbar appeared she used her last breath In pouring out her hatred, and then, when she was out of hearing, carried out her terrible purpose.
Every link Is now In Its place and the chain Is complete” Each convention used and discussed In this essay, demonstrate the style of crime writing written at the time. Closer analysis of salary texts from different periods also begin to show the evolution of the genre over time. The Problem of Thro Bridge By catty case. “l can’t explain things-I can’t even try to explain them, but I know beyond all stonework was grey, but at this one point it showed white for a space not larger than partaking in the incident. This convention is demonstrated in the final paragraph of an arsenal in the house-and kept it for her own use.
A similar one she concealed that morning in Miss Dunbar wardrobe after discharging one barrel, which she could easily do in the woods without attracting attention. She then went down to the bridge where she had contrived this exceedingly ingenious method for getting rid of her weapon. When Miss Dunbar appeared she used her last breath in pouring out her Every link is now in its place and the chain is complete” Each convention used and discussed in this essay, demonstrate the style of crime writing written at the time. Closer analysis of similar texts from different periods also

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More Than One Billion Indians: a Gigantic Problem or a Sea of Opportunities

If the 2011 census is to be believed the current population of India is around 1. 21 billion plus or minus a few millions and we all know to which side the scale is going to tip. The Indian population is varied not only on the lines of religion, region but also on the basis of thought process. While on one hand we are facing the problem of land acquisition from the poorest of poor farmers, on the other hand luxury car maker brand Mercedes has shown a sales growth of 67 percent.
The huge population of India provides a large service sector which plays a crucial role in developing the economy of a nation. Moreover, 30% of the population is between the ages of 10-24 years which clearly shows the amount of untapped energy that India has. If used and guides correctly this can be used to boost our industrial and structural sectors.
The brain drain that India has suffered since last two decades clearly shows that Indians have huge potential and they are just looking for the right niche to develop themselves. Indians have raised themselves from all walks of life and made a mark in fields from astronomy to yoga. On the other hand, the resources that the country has are limited and therefore a burgeoning population restricts the per capita availability. We have seen an increase in the crime rate due to the unavailability of even the most basic amenities.

The government has clearly hit the mark by creating employment opportunities for the rural masses through programs such as NREGA, NRLM, etc. But even creating such employment opportunities costs the government a lot of money and resources. The glass can be half empty or half full. We may say we have more than a billion mouths to feed but we forget that we also have more than 2 billion hands to make such a humongous task possible.