IntMath Newsletter: Graph mystery, science GIFs

29 Mar 2017

In this Newsletter:

1. What's the correct graph of sec(arccos(x))?
2. Resource: Science GIFs
3. Math in the news: income inequality
4. Math movie: Benjamin Banneker
5. Math puzzles
6. Final thought: Priorities

1. What's the correct graph of sec(arccos(x))?

Here we solve another graph mystery - what is the correct domain for y = sec(arccos(x))? And why do different software packages show similar, but different graphs? What's the correct graph of sec(arccos(x))?

2. Resource: Science GIFs

Here's an interesting collection of animated GIFs showing a range of science concepts. I especially liked "Formation of a Nor'easter" and "Wingtip Vortices". See: Science gifs

3. Math in the news - curbing income inequality

There's been discussion in the UK about imposing a maximum wage on top executives as a multiple of the wage of the lowest paid employee. According to Pay caps are neither economically crazy nor morally wrong in the Irish Times:

In the private sector in the US the average chief executive is paid more than 500 times the pay of the average blue-collar worker.

Some countries are considering a Universal Basic Income (e.g. Finland), while others agonize over whether to ensure workers get at least a basic minimum wage.

So what wage are we really worth? One commonly used model of wages as a function of the amount of education and experience is the Mincer Earnings Function, expressed as:

where

w = earnings

w₀ = earnings of someone with no education nor experiences

s = years of schooling

x = years of potential labor market experience

ρ is the return due to schooling

β₁ and β₂ are the returns due to experience

So next time you go to your boss to ask for a raise, take this equation with you.

4. Math movie: The exceptional life of Benjamin Banneker

Self-taught mathematician and astronomer Benjamin Banneker excelled despite little formal education and prejudice. Math movie: The exceptional life of Benjamin Banneker

5. Math puzzles

The puzzle in the last IntMath Newsletter asked about the dimensions and volumes of 2 boxes made from a square sheet of metal.

To make a square open box, we need to cut smaller (green) squares of side s, like this:

In terms of v, the dimensions of the original square are

In terms of s, the dimensions of the original square are

Some of your answers started with a square of side x and found its side length to be:

or

both of which of course are the same as x = 3s.

Correct answers for this part were given by Jatin, Saikrishna, Jeel, Don, Don A, and Chris.

For the bonus part, we need to maximise the length of each of the cube sides, while having enough material left over for the top.

Solution 1

Here's one way to do it, where we cut out the 4 green triangles. The smaller triangular tips are then folded in to create the top.

To find the size of those triangles that need to be cut out, and the final volume, we proceed as follows.

Let the final cube dimensions be m × m × m.

We find distance JK by observing OJK is a right isosceles triangle with equl sides m. Using Pythagoras, we find

We also have (so the triangle tips fold in and fit properly)

So the total length

But we know that length

So

This gives

So the volume of our maximized cube is

That is, it's 1.193 times the volume of our open box.

Saikrishna suggested the above solution.

But does that give us the maximum volume? There's a lot of unused metal left over (the green triangle parts).

Solution 2 - use all the metal

The following approach (suggested by some of you) uses all of the metal sheet. We know we need 6 squares to make the closed box, and thosse squares will have 1/6 the area of the metal sheet.

So if we let the original sheet be 3 × 3 = 9 cm², (for convenience), then each of the 6 squares making the box has equal area 9/6 = 1.5 cm².

This means the side of each square is √1.5 = 1.224745.

So the volume of the box is given by:

That is, our closed box using all the metal will have volume 1.87 times the volume of the first open box.

Here's how we would need to cut the sheet. The 4 squares shown have dimensions √1.5 × √1.5.

Next, I've cut out section above KQ and placed it above rectangle QRIG. You can see it's not quite big enough (the height is 1.1 cm, not 1.2247.)

Similarly, I cut out the rectangle to the right of HS and placed it to the right of SFLR. It's also not quite big enough.

So we would need to cut strips off the remaining square RLCI to make actual squares. This would be quite troublesome, since there would be several little strips to weld together.

While Solution 2 does maximise the volume, from a manufacturing point of view, the most likely approach adopted by a factory would be to use slightly smaller squares.

Solution 3 - use most of the metal

If we use sides of dimension 1.2 cm, it works nicely, with minimal cutting involved and yielding a volume of 1.728 cm³) and only a small amount of left over metal.

The best mathematical solution is not always the most practical answer.

New math puzzle: Integers

For integers m and n, how many solutions (m, n) are there such that m² + n² ≤ 12?

You can leave your responses here.

6. Final thought: Priorities

There is a flurry of excitement regarding Elon Musk's plans to send people to Mars.

But I don't get it.

Don't get me wrong - I'm totally in favor of scientific exploration, but I can't help but think it would be far wiser to spend the considerable sums involved on fixing more pressing problems here on Earth, like climate change, improving educational opportunities, ramping up the water supply and improving its quality, and reducing air pollution, poverty and income inequality.

Now that would leave a more meaningful and lasting legacy, wouldn't it?

Until next time, enjoy whatever you learn.

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