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 over land 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

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This is a min-max calculus problem, where we want to minimize the cost function:

We need a drawing of the situation:
see https://docs.google.com/drawings/d/1PvkU…

where R is the refinery, 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 over land + 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 get
dC/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

...History of DifferentialCalculus
Universidad Iberoamericana
September 20, 2013
Ever since men felt the need to count, the history of calculus begins, which together with Mathematics is one of the oldest and most useful science. Since men felt that need for counting objects, this need led to the creation of systems that allowed them to maintain control of their properties. They initially did it with the use of fingers, legs, or stones. But as humans continued developing intellectually, they achieved to implement systems or more advanced forms that allowed them to solve problems.
The Egyptians were the first civilization to develop mathematical knowledge. They devised numeral systems through hieroglyphs, representing the numbers 1, 10 and 100 through sticks and human figures. This system evolved into what we know today as the Roman system. Other important civilizations in history, such as the Babylonians, created other numeral systems, where the solution to the problem of counting the objects was solved with the implementation of a sexagesimal method. Civilizations as ancient China and ancient India used a hieroglyph decimal system, with the characteristic that these implemented the number zero.
The progress achieved ever since each culture implemented their numeral system are still used today. The algebraic advance of the Egyptians resulted in the...

...increasing |
By the First Derivative Test, x = 4 is a local minimum.
2) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -3 and 6 are the only critical numbers of f.
Testing:
x < -3 | f'(-10) = 672 | f is increasing |
-3 < x < 6 | f(0) = -108 | f is decreasing |
x >6 | f(10) = 312 | f is increasing |
By the First Derivative Test, x = -3 is a local minimum and x = 6 is a local maximum.
3) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', 0 and 0.75 are the only critical numbers of f.
Testing:
x < 0 | f'(-1) = -7 | f is decreasing |
0 < x < 0.75 | f(0.5) = -0.25 | f is decreasing |
x >0.75 | f(1) = 1 | f is increasing |
By the First Derivative Test, x = 1 is a local minimum.
4) We are given the function
First, we find the derivative:
We set the derivative equal to 0 and solve:
Since the domain of f is the same as the domain of f', -2 and 0 are the only critical numbers of f.
Testing:
x < -2 | f'(-10) = 80 e- 10 | f is increasing |
-2 < x < 0 | f(-1) = - e- 1 | f is decreasing |
x > 0 | f(1) = 3 e | f is increasing |
By the First Derivative Test, x = - 2 is a local minimum and x = 0...

...and a comprehensive environment where the youth feel that they have a place to turn to where it is they are, is what will really solve the roots problem.
These were just some of the techniques and plans that have been drawn up. Many of them are similar. Where they differ is where to draw the lines as to what is the most productive and effective tool. By examining this aspect we would know where to concentrate our effort and resources. The National Crime Prevention Council is a leading authority on this issue, although they are only known to most people by the group that sponsors McGruff the crime dog. What does work, works on three different levels, identifying problems in an individual, engaging awareness of the issue, and promoting physical prevention features such as metal detectors. So what works is a balance of Teachers, students, principles, parents, Law enforcement, and the community engaging in the three levels mentioned previously. According to NCPC there is a direct correlation between in community involvement in crime prevention and reported teen crimes.
As far as this issue is concerned the solution to the problem will be identifying the root of the problem. Some people are quick to blame the media, while some blame parents, and others blame the schools themselves. The solution does depend on the problem and identifying is very critical. At...

...teenage girls of their personal image and appearance. This includes but is not limited to their weight, hair color, breast size, color of skin or simply how much makeup they apply. The body type portrayed in advertising as the ideal is possessed naturally by only 5% of American females, yet 47% of girls ages 11-18 grade reported wanting to lose weight because of magazine pictures and 69% of girls reported that magazine pictures influenced their idea of a perfect body shape (Mellin) . This distorted and incorrect view that young girls have of themselves have caused many to develop serious self-confidence issues and eating disorders such as anorexia and bulimia nervosa. This is a tough problem to find solutions to, but by getting the media to use more diverse actresses and models this problem may be helped. By magazines advertising the same tall, skinny, dark skinned woman on their covers, adolescent girls are pressured to feel the need to have to look this way as well. In our society today it can be extremely hard to feel accepted, but by forcing the media to change their close minded opinions, views and advertisements, we can not only change teenage girl’s lives but in some extremely severe cases save them as well.
In a world where 90% of every female celebrity is on the verge of being vastly underweight it is easy for young teenagers to feel pressured to be similar. Being overweight as a young girl can cause isolation, verbal...

...When looking around the world today at our global civilization there are many problems that leap to our attention and everyone has a vested interest, one such example is food. Science has been tinkering with nature for the past one-hundred years trying to come up with solutions to help fight famine, and as a result, man made products have created ever more problems. The easiest way in which an individual or individuals can create change is to effect change in their local communities, so others can see an image of what the world could be. In my local area of Peoria County the topic of food is major problem and all of the issues that correspond, like genetically modified organisms (GM), pesticides, herbicides, cost of food, health of people, and the health of our environment. A practical solution to all these enormous problems we face when we walk through a grocery store and around our community can be fixed through the practice of local organic farming and community participation. With the help of volunteers in Peoria County we could begin growing organic produce that will help create, sustain, connect, and educate our community to help the less fortunate and build a healthier, stronger community.
Experts say that nearly sixty to seventy percent of processed foods contain genetically modified ingredients (Biotech). GM foods are created using biotechnology to change their genetic material....

...student the student to do well, and without putting too much pressure on him/her.
Procedures
These are the procedures I have taken in order to find the solution to this problem:
1. Research
I have created a poll for students in high school to take (30 people), and these were the results:
Are parents and teachers putting too much pressure on teenagers?
Yes, they should encourage the student without overwhelming them will all the work
15 People
50%
Yes, but it’s good preparation for college
10 People
30%
No, there isn’t any pressure
5 People
20%
Articles about students that have commit suicide because of so much pressure:
a. About 40 students commit suicide every year because of pressure during examinations
Stress levels with high amounts of work
b. One in every five high school students are constantly stressed due to too much work
c. Movie- “Race to Nowhere” is a film about the high stress levels students are faced with at school everyday
d. Most school districts have now came up with the fair rule that each class should give out a reasonable amount of half an hour of homework, no more than that, in order for students to be less stressed about high amounts of work, and to give students some free time for themselves.
2. Ideas for how I can change this/inform adults and teachers about this problem:
Poster
a. At first I had the idea to make a few posters and hang them up around the school while parents...

...zones it just seems like none of them work. The problem is that Findlay is getting more rain than the Blanchard River can hold water, which is making the city of Findlay flood. In my opinion we have a couple of choices of ways to get around ruining our homes and lives, which would be corralling the rain to different location, creating more reservoirs, and probably the best solution, building floodwalls. As a community we need to look at all our options and come up with a plan.
The first flood we hear was in 1917, however the first one they could document the levels and severness was back in 1959 and the community has the opinion that since no one has done anything thus far, nothing will be done. As Jackie Fuller explained in her article “Findlay, Your Flood Problem can be Solved” stated, “Over the last two years Findlay, Ohio has flooded 10 times. While flooding is not a new problem, the frequency of it is. There is a solution for the city’s flooding problem. With the help of the Northwest Ohio Flood Mitigation Partnership, the solution will be found and implemented.”(para. 1) Fuller went on to explains that as the weather changes its pattern so will the rainfall, as a result Findlay may still flood, however it won’t be as regular (Para. 2-6). As a community we can hope the weather changes its pattern very soon!
One possible solution to the flooding of...

...Problem Set on Chapter 9.
CAPM, beta, and WACC
1. Bradshaw Steel has a capital structure with 30 percent debt (all long-term bonds) and 70 percent common equity. The yield to maturity on the company’s long-term bonds is 8 percent, and the firm estimates that its overall composite WACC is 10 percent. The risk-free rate of interest is 5.5 percent, the market risk premium is 5 percent, and the company’s tax rate is 40 percent. Bradshaw uses the CAPM to determine its cost of equity. What is the beta on Bradshaw’s stock?
Beta risk
2. Sun State Mining Inc., an all-equity firm, is considering the formation of a new division that will increase the assets of the firm by 50 percent. Sun State currently has a required rate of return of 18 percent, U. S. Treasury bonds yield 7 percent, and the market risk premium is 5 percent. If Sun State wants to reduce its required rate of return to 16 percent, what is the maximum beta coefficient the new division could have?
Risk-adjusted discount rate
3. Assume you are the director of capital budgeting for an all-equity firm. The firm’s current cost of equity is 16 percent; the risk-free rate is 10 percent; and the market risk premium is 5 percent. You are considering a new project that has 50 percent more beta risk than your firm’s assets currently have, that is, its beta is 50 percent larger than the firm’s existing beta. The expected return on the new project is 18 percent. Should the project be accepted if...

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