ERJ Brainteaser: December

Find the connection between these cities (ordered alphabetically) and fill in the missing two.

Antwerp, ___, Barcelona, ___, Melbourne, Montreal, Munich, St. Louis, Sydney. 

Answer: After the year we’ve all had, it is not surprising that many readers have turned their minds to more solid (and, perhaps. liquid) matters. So it is extra well done to last-Brainiacs-standing Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; and John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK. They identified that these, along with Atlanta and Los Angeles, are non-capital cities to have hosted the summer Olympics.

Wishing all our readers - Brainics and non-Brainiacs alike - a very Happy Christmas and a great New Year!

Next teaser in early January, when we will also announce the winners in the prestigious Brainiac of the Year Awards.
 

At a tennis ball manufacturing company, materials scientists are testing their latest product range for bounce.
In one experiment, a ball is thrown vertically upwards with a speed of 16?m?s?1 from a point 80?metres above the ground.
The test guys have to work out the speed with which the ball strikes the ground as well as the total time the ball is more than 85?metres above the ground.
When it lands, the ball rebounds with 20% of the speed with which it strikes the ground. The team then try to work out the greatest height reached by the ball when it rebounds.

What should be the results of the three tests?l

Answer: This really was one of our tougher teasers, so hats off to everyone who had ago. And, extremely well done to the following ace readers, who skilfully worked out the answers at: around 43m?s?1, 2.6s, and 3.7m (solutions below): Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria;  John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; and Michele Girardi, Scame  Mastaf Spa, Suisio, Italy.

Solutions

John Bowen
Mr Newton comes in handy here, as we need to use the equations of motion where acceleration due to gravity is taken as 9.81ms-2
To calculate the speed the ball hits the ground, we need to calculate the initial height above the 80m start:
v.sqd-u.sqd =2fs: at the top velocity = 0, so 16 sqd = 2fs; 256 = 2*9.81*s = 13.05m.
We use this equation again downwards to the ground to calculate velocity at impact: v. sqd = 2fs
v.sqd = 2*9.81*93.05 [total fall is 80 + 13..05 above] = 1825.64
v = 42.73ms-1 This is the velocity with which the ball hits the ground
Use the same equation to calculate the rebound height: vsqd - usqd = 2fs; u = 8.55 [20% of impact speed], v = 0, f = -9.81
73.1 = 2*9.81*s ; s = 3.73m
To calculate the time above 85metres we use the Equation of motion s = ut + 1/2 f.tsqd and solve the resulting quadratic equation using the 80m starting point and height [s] of 5 m:
5 = 16t - 1/2*9.81* tsqd
Rearranging, 4.9*tsqd -16t + 5 = 0
Applying the quadratic formula  [x = [-b+/-{sq rt [bsqd - 4ac]}/2a]
t = 16 +/-SqRt[256-98]/9.8 = [16+/- 12.6]/9.8
t = 0.35 [upwards] and 2.92 [downwards]
So time above 85m high = 2.57 seconds

Michele Girardi
Y=Y0+V0*t-1/2.g.t^
Y0=80
g=10
 V0=16
Y=80+16*t-5*t^2
dY/dt = 16-10*t
Using an excel file and the solver function to find Y equal to  85,85,0
point         t       Y   dY/dt
start       0,00   80,00   16,00
cross 85    0,35   85,00   12,49
max         1,60   92,80    0,00
cross 85    2,85   85,00  -12,49
ground      5,91    0,00  -43,08
rebound V0' = 43.08*0,2 =   8,62
after rebound
V^2=V0'^2-2.g*Y
V^2=8.62^2-20*Y
Y  =(0-8.62^2)/(-20)
max          Y=     3.72m
Speed when the ball hits the ground : 43.1 m/s
Time above 85 m  : 2.85-0,35 =    2.5s
Heigth of rebound           3.72 m
 

Bonus question

What should the values of X be in the third row of this table?
 

The key to the equally tricky Bonus question was to identify that adding the numbers in the first two squares if each row (right to left) and then adding the digits to get the next number, ie in the third column. So, in the fourth row, 8+8=16, 1+6 =7 and 8+7=15, 1+5=6. Very well done here to: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; and Mehmet Koral, Erhardt-Leimer representative for Turkey, managing director, C&C Endüstriyel Dan??manl?k, E?itim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey.

New teaser on Monday
 

Which country comes next in this series

Belgium, Bulgaria, Cyprus, Denmark, Germany, _ , ….

Answer: Only one correct reply to this week’s main teaser. So extra congratulations to Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria, who quickly worked out: “The answer is Hungary. (13th position in the alphabetical list of EU countries, applying prime numbers).”

From this diagram of a regular pentagon inside a square, can you work out the value of X?

Bonus question

Which two numbers do not belong in this list of two-digit numbers?
30, 31, 32, 33, 34, 36, 39, 40, 43, 44, 45, 46, 47, 48, 49…

Answer: This one tripped up quite a few, with only a select set of readers correctly working out the answer at 54° (neat solutions below). Very well done to: Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Michele Girardi, Scame  Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Francesco Parazzoli, R&D -  material development, new raw materials, Pirelli Tyre SpA, Milan, Italy;  Mehmet Koral, Erhardt-Leimer representative for Turkey, managing director, C&C Endüstriyel Dan??manl?k, E?itim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey.

There was only one correct reply to the Bonus question. First thing Monday, Stephan Paischer identified that 44 and 47 don’t belong, as they are phone prefixes on non-EU countries (UK and Norway). All the other are phone prefixes of EU countries.

Solutions to main Pentagon question

:1)    The base angle of a Pentagram is 108°.
2)    The opposing angle is 72° (180-108).
3)    The angles of the triangle on the down right side are 72+90+y=180°, y being therefore 18°.
4)    The requested angle x is: 18+108+x=180°, x being therefore 54°.
Stephan Paischer 

Value of angle X = 54 deg [180 - 108 [internal angle of pentagon ]-18[remaining angle of triangle]
John Bowen

Let's call  O the center of the pentagon, A the top vertex and  B the origin of X
- AOB =  360°/5 = 72°
- OAB = OBA =  (180-72)/2 = 54°
- X = OAB= 54°, since they are alternate interior angles Cheers
Michele Girardi

Exterior angle is 360/5 = 72 deg.
Interior angle is 180-72 = 108 deg
The angle opposite the exterior angle in the white triangle to the lower right in your diagram is 90 -72 = 18 deg
Angle X therefore is 180 -18 -108 = 54 deg
Andrew Knox

New teaser on Monday