Question

The length of an arc of a circle is 7.5 cm. The corresponding sector area is 37.5 cm?.
Find:
a) the radius of the circle
b) the angle subtended at the centre of the circle by the arc​

Answer 1

Answer:

thank you!!!!

Step-by-step explanation:

Let x represent the angle subtented at the center of the circle:

Area of sector = (x/360)*pi*r²

37.5 = (x/360)* pi * r²

solve for r²:

37.5/(x/360 * pi) = r²  ---> equation 1

arc length = (x/360)*pi*2r

7.5 = (x/360)*pi * 2r

solve for r:

7.5/(x/360 * pi) = 2r

3.75/(x/360 * pi) = r

square both sides:

(3.75)²/(x/360 * pi)² = r²  ---> equation 2

since both these equations = r², we can set them equal to each other:

(3.75)²/(x/360 * pi)² = 37.5/(x/360 * pi)

cross multiply:

(3.75)²(x/360 * pi) = 37.5(x/360 * pi)²

divide both sides by (x/360 * pi)

(3.75)² = 37.5(x/360 * pi)

14.0625 = [(37.5)(3.14)/360]x

14.0625 = 0.32725 x

42.97° = x

and:

7.5 = (x/360)*pi * 2r

7.5 = 42.97/360 * pi * 2r

7.5 = .75 r

10 = r

radius = 10 cm

Central angle = 42.97°

Answer 2

Answer:

a) The radius of circle is 10 cm.

b) The angle subtended by arc at centre is 0.75 radians.

Step-by-step-explanation:

We have given that,

Length of arc of circle = 7.5 cm

Area of corresponding sector = 37.5 cm²

Let the radius of circle be r cm.

And the angle subtended at centre of circle by arc be θ radians.

We know that,

Length of arc = r θ

⇒ 7.5 = r θ

⇒ θ = 7.5 / r - - - ( 1 )

We know that,

Area of sector = 1 / 2 * r² θ

⇒ 37.5 = 1 / 2 * r² * 7.5 / r

⇒ 37.5 = r² / r * 7.5 / 2

⇒ 37.5 = r * 7.5 / 2

⇒ r = 37.5 * 2 / 7.5

⇒ r = 5 * 2

⇒ r = 10 cm

Now,

θ = 7.5 / r - - - ( 1 )

⇒ θ = 7.5 / 10

⇒ θ = 0.75 radians

∴ The radius of circle is 10 cm and angle subtended by arc at centre is 0.75 radians.