Question

See the attachments below.

Answer 1

Answers:

a) $L=6cm$

b) $A=\frac{3\pi}{2}$

Step-by-step explanation:

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

$A=\frac{rL}{2}$ (1)

and

$A=\frac{r^{2}\theta}{2}$ (2)

Where:

$A=27cm^{2}$

$r$ is the radius

$L=\frac{4}{x}cm$ (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):

$27cm^{2}=\frac{r(\frac{4}{x}cm)}{2}$ (4)

Isolating $x$:

$x=\frac{2r}{27cm}$ (5)

Substituting (5) in (3):

$L=\frac{4}{\frac{2r}{27cm}}cm$  (6)

Solving:

$L=\frac{54cm^{2}}{r}$  (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):

$A=\frac{rL}{2}=\frac{r^{2}x}{2}$ (8)

Isolating $r$:

$r=\frac{L}{x}$ (9)

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

$r=\frac{\frac{54cm^{2}}{r}}{\frac{2r}{27cm}}$ (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 $L$

$L=\frac{54cm^{2}}{9cm}$ (11)

$L=6cm$ (12)

b) In this second case we have:

$L=S$ is the length of arc

$\theta=\frac{\pi}{3}$ 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):

$A=\frac{rL}{2}=\frac{r^{2}\theta}{2}$ (13)

$\frac{3S}{2}=\frac{3^{2}(\frac{\pi}{3})}{2}$ (14)

$3S=9\frac{\pi}{3}$ (15)

$S=\pi$ (16)

Knowing $A=\frac{3S}{2}$:

$A=\frac{3\pi}{2}$ This is the area of the sector of the circumference.