ERJ Brainteaser - July
15 Jul 2024
Each month, ERJ sets a weekly brainteaser, with questions of varying degrees of difficulty. Readers supplying the most accurate (and stylish) answers are then considered for the prestigious Brainiac of the Month title.
Question 2: Come fly with me
Materials scientist Gail’s plane from London to Edinburgh takes off at exactly the same time as colleague David’s flight departs Edinburgh for London. A wind current of around 5mph is travelling against the David’s plane. Both planes are travelling at 120 miles per hour relative to the wind (not the ground). If the flight distance is 390 miles each way, how long will it take for the two planes to pass each other mid-flight?
Email your answer: correct replies on Friday.
Question 1: Two missing
31 |
13 |
18 |
18 |
47 |
16 |
63 |
31 |
54 |
28 |
26 |
26 |
16 |
45 |
61 |
29 |
25 |
? |
86 |
? |
47 |
8 |
39 |
39 |
Answer: Ironically, all but two of our Brainiacs were missing when it came to answering this tricky teaser. So, it’s extremely well done to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; and Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA.
SOLUTION
Kamila Staszewska
The missing numbers are 61 and 36
Number in second column is difference between number in first and third column: 86-25 = 61
Number in fourth column is difference between number in first and second column: 61-25 = 36
The other numbers can be worked out as follows
A |
B |
C |
D |
1 |
b+c |
a-c |
D |
a-b |
2 |
b+d |
a-c |
a+b |
a-b |
3 |
b+c |
a-c |
D |
a-b |
4 |
b+d |
a-c |
a+b |
a-b |
5 |
b-d |
a-c |
a+b |
a-b |
6 |
b+c |
a-c |
D |
a-b |
New teaser on Monday.
Question 1: Larry’s lawn
Landscaper Larry has designed a lawn represented by the triangle ABC, shown in the figure. If length AB is to be 30m long, angle BAC = 70° and angle ABC = 60°, what is the approximate area of the lawn?
Answer: Trigonometry was the key to this tricky gardening teaser, which generated impressive replies (see Solutions below) from: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Wong SiauWoon, R&D manager, gloves company, Malaysia; John Bowen, consultant, Bromsgrove, UK. Very well done to all above and everyone else who had a go.
SOLUTIONS
Andrew Knox
Draw a line perpendicular to the base intersecting at C.
Then height h = 30 / (tan 20 + tan 30) = 31.87 m.
Area = (30 x 31.87) / 2 = 478 m2.
Kamila Staszewska
Area=BC * h/2 ; h=?, BC=?
sin(<ABD)=h/AB ; <ABD=60°
sin(60°)=(√3)/2=h/30
h=√3*15=25.981m
cos(<ABD)=BD/AB
cos(60°)=0.5=BD/30
BD=0.5*30=15m
tan(<CAD)=CD/h ; <CAD=40°
tan(40°)=0.8390996=CD/25.981
CD=0.890996*25.981=21.801m
BC=BD+CD=15+21.801=36.801m
Area = BC*h/2=36.801*25.981/2 = 478.06m^{2}^{}^{}
Sudi Sudarshan
Drop a perpendicular CD from vertex C to base AB.
Let AD = x, then DB = (30-x)
Tan(^A) = CD/AD
Tan(70°) = h/x
2.747 = h/x or h = 2.747x
Tan(^B) = CD/DB
Tan(60°) = h/(30-x)
1.732 = h/(30-x)
h = 1.732(30-x)
2.747x = 1.732(30-x)
x = 1.732*30/4.479 = 11.600
h = 2.747x = 2.747*11.600 = 31.873
Area of triangle = 0.5*b*h
= 0.5*30*31.873 = 478.095
Wong SiauWoon
Angle CAB = 70
angle CBA = 60
angle ACB = 180 - 70 - 60 = 50
30/sin C = BC/sin70
30/sin50 = BC/sin70
BC = (30/sin50)*sin70
BC = (30/0.766) * 0.940
BC = 39.165 * 0.940
BC = 36.82m
Area = 1/2(30*36.82)sin60
Area = 1/2(1104.6)(0.866)
Area = 478.3 m2
John Bowen
The area of any triangle is given by Area = 1/2x base x height
We have a triangle, base 30 metres and height X metres
We need to calculate X, which we do by dividing the lawn into 2 triangles of 70/20/90 degrees, height X and 60/30/90 degrees, height X
The lengths of the two sides opposite the 20 and 30 degrees we shall call a and b, such that [a + b] = 30 [we know this because you told us]
We can now set to with some trigonometry:
Tan 20 = a/X = 0.364, and Tan 30 = [30 - a]/X = 0.577 [from tables] and rearranging:
a = 0.364X and
30 - a = 0.577X
Adding these together: 30 = 0.941X
X = 31.88 metres
so the area is 1/2 x 30 x 31.88 = 478.2 square metres.
Regards, [and credit to my old maths master].
Michele Girardi