Question

OAB is a sector of a circle with centre O and radius 8 cm.

AÔB = xº.
(a) Write down an expression, in terms of x and ,
for the area of the sector OAB.
(b) PQR is a semicircle of radius 4 cm.
The area of the sector OAB is of the area of
this semicircle.
Calculate the value of x.

Answer 1

The sector's area is given by the expression (64π)(x/360). Comparing this to a semicircle with radius 4 cm, solving the equation (64π)(x/360) = (8π)/3 leads to x being 54 degrees.

Step-by-step explanation:

The student is dealing with the topic of finding the area of a sector in a circle and comparing it to the area of a semicircle.

Solution for (a)

The formula for the area of a sector, given by angle x degrees in a circle of radius r, is A = (πr²)(x/360).

For a circle with radius 8 cm, the expression for the area of the sector OAB is

A = (π*8²)(x/360)

= (64π)(x/360).

Solution for (b)

The area of a semicircle with radius 4 cm is (1/2)π(4²) = 8π.

Given that the area of the sector OAB is one third of the area of the semicircle, we set up the equation

(64π)(x/360) = (8π)/3.

Solving for x we get x = 54 degrees.