Answer
(a.) $34.9 \; cm$
(b.) $349 \; cm^2$
(c.) $152 \; cm^2$.
Work Step by Step
Given values are:-
Radius $r=20.0\; cm$
Central angle $\theta = 100.0^{\circ}$
Length of the cord $c= 30.6 \; cm$.
By using conversion factor $\frac{\pi \; rad}{180^{\circ}}$ central angle in radian measure is
$\theta = 100.0^{\circ}\times \frac{\pi \; rad}{180^{\circ}}$
Simplify.
$\theta = \frac{5\pi }{9}\; rad$
(a.) Formula for the length of the arc is $s=r\theta$.
Substitute all values.
$s=(20.0\;cm)(\frac{5\pi }{9})$
Simplify.
$s=34.9\; cm$ (correct to one decimal place.)
(b.) Formula for the area of the sector is
$A=\frac{1}{2} r^2 \theta $
Substitute all values.
$A=\frac{1}{2} (20.0 \; cm)^2 (\frac{5\pi }{9}) $
Simplify.
$A=349 \;cm^2$ (Rounded value.)
(c.) Formula for the area of the segment is
$A=\frac{1}{2}r^2\theta-\frac{c\sqrt{4r^2-c^2}}{4}$
From the part (b.) $\frac{1}{2} r^2 \theta = 349.1\; cm^2 $
Substitute all values into the formula.
$A=349.1\;cm^2-\frac{(30.6\; cm)\sqrt{4(20.0\;cm)^2-(30.6\; cm)^2}}{4}$
Simplify.
$A=349.1\;cm^2-197.07$.
$A=152 \;cm^2$ (Rounded value.)
