Question

Help me , it is 50 point ​

Answer 1

Answer:

a) 52.0 cm

b) 62.8 cm

c) 114.8 cm

Conversion:

120° = 2π/3 radians

(a) Find Length of chord PR;

⇒ 2(radius)sin(θ/2)

⇒ 2(30)sin((2π/3)/2)

⇒ 2(30)sin((2(3.142)/3)/2)

⇒ 30√3

⇒ 52.0 cm

(b) Find Length of arc PQR;

⇒ radius(θ)

⇒ 30(2π/3)

⇒ 30(2(3.142)/3)

⇒ 62.8 cm

(c) The perimeter of shaded region;

⇒ Length of chord + Length of arc

⇒ 62.84 + 51.965

⇒ 114.8 cm

Answer 2

Answer:

(a)  52.0 cm

(b)  62.8 cm

(c)  115 cm

Step-by-step explanation:

Part (a)

$\textsf{Chord length}=2r \sin \left(\dfrac{\theta}{2}\right)$

where:

• r = radius
• $\theta$ = central angle (measured in degrees)

Given:

• r = 30 cm
• $\theta$ = 120°
• chord = PR

Substitute the given values into the formula:

$\begin{aligned}\implies PR & =2(30)\sin \left(\dfrac{120^{\circ}}{2}\right)\\\\& = 30\sqrt{3}\\\\ & = 52.0\:\sf cm\:(3\:sf)\end{aligned}$

Part (b)

$\textsf{Arc length}=2 \pi r\left(\dfrac{\theta}{360^{\circ}}\right) \quad$

where:

• r = radius
• $\theta$ = central angle (measured in degrees)

Given:

• r = 30 cm
• $\theta$ = 120°
• arc = PQR
• $\pi$ = 3.142

Substitute the given values into the formula:

$\begin{aligned}\implies PQR & =2 (3.142)(30)\left(\dfrac{120^{\circ}}{360^{\circ}}\right) \quad\\\\ & = \dfrac{1571}{25}\\\\ & = 62.8 \sf \:\:cm \:(3 \:sf)\end{aligned}$

Part (c)

$\begin{aligned}\textsf{Perimeter of shaded portion} & = \textsf{chord length} + \textsf{arc length}\\\\ & = 30\sqrt{3}+\dfrac{1571}{25}\\\\ & = 114.80152...\\\\ & = 115\:\: \sf cm\:(3\:sf)\end{aligned}$