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# areas

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https://vle.mathswatch.co.uk/images/questions/question16574.png

Apr 1, 2018

#1
+8962
+4

Let  27 cm  be the base of triangle CDE .

Let  h cm  be the height.

sin( angle )   =   opposite / hypotenuse

sin 32°  =   h / 14                                         Multiply both sides by  14 .

14 sin 32°   =   h

area of triangle CDE   =   (1/2)(base)(height)

area of triangle CDE   =   (1/2)(27)(14 sin 32°)

area of triangle CDE   =   189 sin 32°

area of triangle CDE   ≈   100.155

By the law of cosines...

EC2   =   142 + 272 - 2(14)(27) cos 32°

EC2   =   196 + 729 - 756 cos 32°

EC2   =   925 - 756 cos 32°

area of sector / area of circle  =  105° / 360°

area of sector   =   105° / 360° * area of circle

area of sector   =   105° / 360° * pi * EC2

area of sector   =   105° / 360° * pi * (925 - 756 cos 32°)

area of sector   ≈   260.115

area of logo   =   area of sector + 2(area of triangle CDE)

area of logo   ≈   260.115 + 2(100.155)

area of logo   ≈   460.425

And  460.425  rounded to the nearest ten   =   460

Apr 1, 2018
edited by hectictar  Apr 1, 2018

#1
+8962
+4

Let  27 cm  be the base of triangle CDE .

Let  h cm  be the height.

sin( angle )   =   opposite / hypotenuse

sin 32°  =   h / 14                                         Multiply both sides by  14 .

14 sin 32°   =   h

area of triangle CDE   =   (1/2)(base)(height)

area of triangle CDE   =   (1/2)(27)(14 sin 32°)

area of triangle CDE   =   189 sin 32°

area of triangle CDE   ≈   100.155

By the law of cosines...

EC2   =   142 + 272 - 2(14)(27) cos 32°

EC2   =   196 + 729 - 756 cos 32°

EC2   =   925 - 756 cos 32°

area of sector / area of circle  =  105° / 360°

area of sector   =   105° / 360° * area of circle

area of sector   =   105° / 360° * pi * EC2

area of sector   =   105° / 360° * pi * (925 - 756 cos 32°)

area of sector   ≈   260.115

area of logo   =   area of sector + 2(area of triangle CDE)

area of logo   ≈   260.115 + 2(100.155)

area of logo   ≈   460.425

And  460.425  rounded to the nearest ten   =   460

hectictar Apr 1, 2018
edited by hectictar  Apr 1, 2018