Short Questions (Question 1 & 2)

Posted on April 21, 2020 by user

Question 1:

The figure shows the sector OCB of radius 13 cm at the centre O. The length of the arc CB = 5.2 cm. Find

(a) the angle in radians,

(b) the perimeter of the shaded region.

Solution:
(a)

$\begin{array}{l} s = r θ \\ 5.2 = 13 (∠ C O B) \\ ∠ C O B = 0.4 radian \end{array}$

(b)

$\begin{array}{l} \cos ∠ C O B = \frac{O A}{O C} \\ \cos 0.4 = \frac{O A}{13} (change calculator to Rad mode) \\ O A = 11.97 cm \\ ∴ A B = 13 - 11.97 = 1.03 cm \\ C A = \sqrt{13^{2} - {11.97}^{2}} \\ C A = 5.07 cm \end{array}$

Perimeter of the shaded region = 5.07 + 1.03 + 5.2 = 11.3 cm.

Question 2:

The figure shows the sector AOB of a circle, centre O and radius 5 cm. The length of the arc AB is 6 cm. Find the area of:

(a) the sector AOB,

(b) the shaded region.

Solution:
(a) Arc AB = 6cm

s = r θ

6 = 5 θ

θ = 6/5 rad

$\begin{array}{l} Area of sector A O B \\ = \frac{1}{2} r^{2} θ = \frac{1}{2} {(5)}^{2} (\frac{6}{5}) = 15 c m^{2} \end{array}$

(b)

$\begin{array}{l} Area of shaded region \\ = \frac{1}{2} r^{2} (θ - \sin θ) (change calculator to Rad mode) \\ = \frac{1}{2} {(5)}^{2} (\frac{6}{5} - \sin \frac{6}{5}) \\ = 3.35 {cm}^{2} \end{array}$