\[\sinh(\ln\varphi)=\frac{1}{2}\]
cherry-merchant:

Sierpinski transformation

cherry-merchant:

Sierpinski transformation

rootbombs:

There’s a door at my school and this was on it

penis-hilton:

givemeinternet:

The closer to the end the more satisfying it gets…

the last one was an orgasm

penis-hilton:

givemeinternet:

The closer to the end the more satisfying it gets…

the last one was an orgasm

Generation Y: the demographic the budget forgot

guardian:

Age: Newborn.

Appearance: A red box of broken dreams and orphans’ tears.

Oh dear, I’m not going to like this, am I? Well, it depends – are you a pensioner?

No. Do you have a job?

I’m applying for lots of them. Do you have thousands of pounds in savings?

Course I bloody don’t. Then no, you’re not going to like it.

Ugh. Lay it on me. It’s a budget “for savers”, and people with savings tend to be old people. In fact, scratch that – given that times are hard all round, people with savings tend to be rich old people.

I’m amazed a Tory chancellor would only cater to the old and rich. Indeed. This budget is an absolute dream for pensioners.

Read More

There are a number of steps to follow when you come across an integration problem that requires trig substitution. The first step is to cry. The second step is to re-evaluate your life and wonder what you did to deserve this.
Professor Leslie Buck (via mathprofessorquotes)
Real analysis is just triangle inequality with applications.
Analysis professor (via mathprofessorquotes)
how to solve this optimization problem ? help?

letmequest:

hi, A manufacturer is planning to sell a new product at the price of $230 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, consumers will buy approximately [ 350y/(y+1) ] + [ 50x/ (x+3) ] units of the product. If manufacturing costs $130 per unit, how much should the manufacturer spend on development and how much on promotion to generate the largest possible profit from the sale of this product? Thanks,

Just to rephrase the question:

Maximise profit given:

  • Price per unit $230
  • Cost per unit $130
  • Research cost $1000\(x\)
  • Promotion cost $1000\(y\)
  • Units sold:

\[\frac{350y}{y+1}+\frac{50x}{x+3}\]

From this information, we can calculate the profit \(P\).

So profit \(P\) = (Price per unit - cost per unit) * (number f units sold) - (research cost + promotion cost).

So,

\[ P = (230 - 130)\times \left( \frac{350y}{y+1}+\frac{50x}{x+3} \right) - 1000(x+y)\]

\[ P = 100\left( \frac{350y}{y+1}+\frac{50x}{x+3} - 10(x+y)\right)\]

\[ P = 1000 \left( \frac{35y}{y+1}+\frac{5x}{x+3} - x - y \right)\]

Hope this is all clear so far. Now we have an expression for the profit, we need to maximise it. We can do this by taking partial derivatives with respect to our variables (in this case \(x\) and \(y\)).

So,

\[P_x = \frac{\partial P}{\partial x} = 1000\left( 0 + \frac{15}{(x+3)^2} - 1 - 0\right) \]

\[P_y = \frac{\partial P}{\partial y} = 1000\left( \frac{35}{(y+1)^2} + 0 - 0 - 1\right) \]

\(P\) is at its maximum when \(P_x=P_y=0\) so,

\[P_x = 0 \Rightarrow 15 = (x+3)^2 \Rightarrow x = \sqrt{15}-3\]

\[P_y = 0 \Rightarrow 35 = (y+3)^2 \Rightarrow y = \sqrt{35}-1\]

So \((x,y)=(1,5)\) to the nearest integer values of \(x,y\).

I hope that helps.

theglammaster:

1: Thou shalt not divide by zero
2: Thou must show thy work
3: Thou must do unto one side of the equation what thou dost unto the other
4: Thou shalt not forget to carry the 1
5: Thou must honour the order of operations
6: Thou must flex the hexaflexagon with caution and never to cause…

aqqindex:

Glenn Sundby, Cairo, 1977

aqqindex:

Glenn Sundby, Cairo, 1977

isomorphismes:

Pythagorean Theorem
This is how I first really understood the Pythagorean Theorem.
The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.
Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.
Ideas involved here:
scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)
circumcircle & incircle
√2
This is also how I first really understood √2, now my favourite number.

isomorphismes:

Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

  • scaling properties of squared quantities
    (gravitational force, skin, paint, loudness, brightness)
  • circumcircle & incircle
  • √2

This is also how I first really understood √2, now my favourite number.

Carl Sagan
183 plays

yellowtypophile:

The Pale Blue Dot

Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives.

The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every “superstar,” every “supreme leader,” every saint and sinner in the history of our species lived there - on a mote of dust suspended in a sunbeam.

The Earth is a very small stage in a vast cosmic arena. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner. How frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds. Think of the rivers of blood spilled by all those generals and emperors so that, in glory and triumph, they could become the momentary masters of a fraction of a dot.

Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves.

The Earth is the only world known so far to harbour life. There is nowhere else, at least in the near future, to which our species could migrate. Visit, yes. Settle, not yet. Like it or not, for the moment the Earth is where we make our stand.

It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.

thegreekletterdelta:

Pokemon as described by physical laws