ERJ Brainteaser - July

ERJ Brainteaser: Two missing

Email your answer: correct replies on Friday.

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 = 478.06m²

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