Answer:
Part A) [tex]A_s=\frac{\pi r^{2}}{360^o}{\theta}[/tex]
Part B) option 1,option 4
Step-by-step explanation:
Part A) Derive the formula for the area of a sector
we know that
The area of circle is equal to
[tex]A=\pi r^{2}[/tex]
The area of circle subtends a central angle of 360 degrees
so
using proportion
Find out the area of a sector [tex]A_s[/tex] by a central angle of ∅ degrees
[tex]\frac{\pi r^{2}}{360^o}=\frac{A_s}{\theta}[/tex]
[tex]A_s=\frac{\pi r^{2}}{360^o}{\theta}[/tex]
Part B) Verify each case
case 1) we have
radius = 5 cm
angle = 120°
area = 26.2 cm 2
Find the area of the sector and then compare with the value of the given area
assume
[tex]\pi=3.14[/tex]
substitute the given values
[tex]A_s=\frac{(3.14)(5)^{2}}{360^o}{120^o}[/tex]
[tex]A_s=26.2\ cm^2[/tex]
so
The given value of area is correct
case 2) we have
radius = 4 cm
angle = 105°
area = 16.7 cm 2
Find the area of the sector and then compare with the value of the given area
assume
[tex]\pi=3.14[/tex]
substitute the given values
[tex]A_s=\frac{(3.14)(4)^{2}}{360^o}{105^o}[/tex]
[tex]A_s=14.7\ cm^2[/tex]
so
The given value of area is not correct
case 3) we have
radius = 6 cm
angle = 85°
area = 23.7 cm 2
Find the area of the sector and then compare with the value of the given area
assume
[tex]\pi=3.14[/tex]
substitute the given values
[tex]A_s=\frac{(3.14)(6)^{2}}{360^o}{85^o}[/tex]
[tex]A_s=26.7\ cm^2[/tex]
so
The given value of area is not correct
case 4) we have
radius = 7
angle = 75°
area = 32.1 cm 2
Find the area of the sector and then compare with the value of the given area
assume
[tex]\pi=3.14[/tex]
substitute the given values
[tex]A_s=\frac{(3.14)(7)^{2}}{360^o}{75^o}[/tex]
[tex]A_s=32.1\ cm^2[/tex]
so
The given value of area is correct
