Question

See the attachments below.

Answer 1

Answers:

a)
`L=6cm`

b)
`A=(3\pi)/(2)`

Explanation:

a) The area of the sector of a circle
`A` is given by:

`A=(rL)/(2)` (1)

and

`A=(r^(2)\theta)/(2)` (2)

Where:

`A=27cm^(2)`

`r` is the radius

`L=(4)/(x)cm` (3) is the length of arc

`\theta=x` is the angle in radians

In this case we have to find the value of
`L`. So, let's begin substituting the known values in (1):

`27cm^(2)=(r((4)/(x)cm))/(2)` (4)

Isolating
`x`:

`x=(2r)/(27cm)` (5)

Substituting (5) in (3):

`L=(4)/((2r)/(27cm))cm` (6)

Solving:

`L=(54cm^(2))/(r)` (7) At this point we have
`L`, but we need to find the value of
`r` in order to have the actual value of the length of arc.

Making (1)=(2):

`A=(rL)/(2)=(r^(2)x)/(2)` (8)

Isolating
`r`:

`r=(L)/(x)` (9)

Substituting (7) and (5) in (9):

`r=((54cm^(2))/(r))/((2r)/(27cm))` (10)

Finding
`r`:

`r=9cm` (10) Now that we have the value of the radius, we can substitute it in (7) and finally find the value of the
`L`

`L=(54cm^(2))/(9cm)` (11)

`L=6cm` (12)

b) In this second case we have:

`L=S` is the length of arc

`\theta=(\pi)/(3)` is the angle in radians

`r=3` the radius

We have to find the area of the sector
`A` and we will use equations (1) and (2):

`A=(rL)/(2)=(r^(2)\theta)/(2)` (13)

`(3S)/(2)=(3^(2)((\pi)/(3)))/(2)` (14)

`3S=9(\pi)/(3)` (15)

`S=\pi` (16)

Knowing
`A=(3S)/(2)`:

`A=(3\pi)/(2)` This is the area of the sector of the circumference.