Week 5Supernovae

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Volume of a sphere

You might not have covered this yet, but there is an equation that that gives the volume of a sphere based on the radius. Just like there is an equation that gives the circumference of a circle based on the radius (circumference = 2×π×r). This short video reveals that equation.

So, now that you know that, try and answer the following questions. Remember:

1 mark

1.1. What is the volume of a sphere with a radius of 2 cm?

  • 8.4 cm3
  • 18.8 cm3
  • 25.1 cm3
  • 33.5 cm3

Volume =43×π×23 =323×π=33.5 cm3.

2 marks

1.2. What is the volume of a sphere with a radius of 8.4 cm?

(Give your answer accurate to the nearest whole number. And use an accurate value of π, using several decimal places.)

Correct Solution: 2,483 cm3

Volume =43×π×8.43=43×592.70×π=2,482.7=2,483 cm3.

2 marks

1.3. What is the approximate volume of a sphere with a radius of 0.909 cm?

  • 1/π cm3
  • π cm3
  • 2π cm3
  • π3 cm3

Volume =43×π×0.9093=43×π×34=π cm3.

3 marks

1.4. In the video, is the speaker Allison Moffett left-handed or right-handed? You can’t be sure, but what is your best guess?

  • Left
  • Right

At first sight, Allison seems to be left-handed, but look at the writing on the glass: Allisson is writing on the wrong side of the glass, and yet her scribbles make perfect sense. I would expect her to either write very slowly, or make lots of mistakes, because it appears that Allison is writing back-to-front on her side, so it looks correct to us.

Another explanation is that Allison is right-handed, that her writing in reality is completely back-to-front from our point of view, but then the film editor flipped the video so that the writing makes sense to us... and as a side effect Allison now appears to be left-handed, while she is actually right-handed.

2. Supernovae explained

Take a look at this very short Open University video, which is all about what happens when a star explodes.

2 marks

2.1 The universe began with the lightest elements in the Periodic Table. The video explains that all the heavier atoms were made when stars exploded, so everything around us is made of stardust, including the calcium in our bones and the iron in our blood. So you and I are made of stardust!

There are other ways to express this - which one of the following statements is basically true?

  • We are made of nuclear waste.
  • We are built from rainbows and unicorn breath.
  • Our nuclear origin means we all have super powers.
  • Our bodies are powered by nuclear reactions.

Some of the elements in our body, namely carbon, nitrogen and oxygen, were created by nuclear reactions that take place before stars explode. However, the heavier elements, such as calcium, iron, phosphorous and zinc in our body, were left behind after nuclear reactions took place during supernovae explosions. So, yes, we are basically nuclear waste... or stardust.

3. Junior Maths Challenge Problem

3 marks

3.1 Ulysses, Kim, Mei, and Tanika had their 12th, 14th, 15th and 15th birthdays in 2013. In what year will their ages first total 100?

  • 2023
  • 2024
  • 2025
  • 2057
  • 2113

In 2013, the total of their ages was 12+14+15+15=56.

This needs to increase by 10056=44 years in order to reach 100 years. Each year they all grow 1 year older, and so the total of their ages increases by 4. This will take 444=11 years. In 11 years time it will be 2013+11=2024.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelogams.

Cheerio, Simon.