Given the dartboard of diameter [tex]20in[/tex], divided into 20 congruent sectors,
The area of a circle is given by the formula
[tex]A=\pi r^2[/tex]
A sector of a circle is a fraction of a circle. The fraction is given by [tex]\frac{\theta}{360^\circ}[/tex]. Where [tex]\theta[/tex] is the angle subtended by the sector at the center of the circle.
The formula for computing the area of a sector, given the angle at the center is
[tex]A_s=\dfrac{\theta}{360^\circ}\times \pi r^2[/tex]
We given a circle (the dartboard) with diameter of [tex]20in[/tex], divided into 20 equal(or, congruent) sectors
To find the central angle, divide [tex]360^\circ[/tex] by the number of sectors. Let [tex]\alpha[/tex] denote the central angle, then
[tex]\alpha=\dfrac{360^\circ}{20}\\\\\alpha=18^\circ[/tex]
The fraction of the circle that one sector takes up is found by dividing the angle a sector takes up by [tex]360^\circ[/tex]. The angle has already been computed in Part I (the central angle, [tex]\alpha[/tex]). The fraction is
[tex]f=\dfrac{\alpha}{360^\circ}\\\\f=\dfrac{18^\circ}{360^\circ}=\dfrac{1}{20}[/tex]
The area of one sector can be gotten by multiplying the fraction gotten from Part II, with the area formula. That is
[tex]A_s=f\times \pi r^2\\=\dfrac{1}{20}\times3.14\times\left(\dfrac{20}{2}\right)^2\\\\=\dfrac{1}{20}\times3.14\times10^2=15.7in^2[/tex]
Learn more about sectors of a circle https://brainly.com/question/3432053
