Answer:
Step-by-step explanation:
From the picture attached,
Area of ΔECD = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]
= [tex]\frac{1}{2}(EF)(CD)[/tex]
From ΔEFD,
sin(32°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
sin(32°) = [tex]\frac{EF}{ED}[/tex]
EF = 14 × sin(32°)
= 7.42 cm
By cosine rule,
EC² = DE² + CD² - 2(DE)(CD)cos(32°)
EC² = 14² + 27² - 2(14)(27)cos(32°)
EC² = 196 + 729 - 641.12
EC² = 283.88
EC = 16.85 cm
Area of ΔECD = Area of ΔAEB = [tex]\frac{1}{2}(7.42)(27)[/tex]
= 100.17
Area of ΔECD + Area of ΔAEB = 2(100.17)
= 200.34 cm²
Area of sector BEC = [tex]\frac{\theta}{360}(\pi r^{2})[/tex]
Here, θ = central angle of the sector
Area of sector BEC = [tex]\frac{105}{360}(\pi)( EC)^{2}[/tex]
= [tex]\frac{105\pi}{360}(16.85)^2[/tex]
= 260.16 cm²
Area of the logo = Area of triangles AEB + Area of triangle ECD + Area of sector BEC
= 200.34 + 260.16
= 460.50
≈ 460 cm²
