Question

A sector of a circle has perimeter `32` `cm` and area `63` `cm^2`. What is the radius length and the magnitude of the angle subtended at the centre of the two possible sectors?

Answer 1

Radius is 11.22cm
Magnitude of angles are 0.85rad or 5.43 rad

Explanation:

length of arc (L)=`rtheta`
area of sector (A) = `1/2r^2theta`

Perimeter = 32
`rtheta+r+r=32`
`rtheta+2r=32`
`r(theta+2)=32`
`r=32/(theta+2)` --- (1)

Area of sector = 63
`1/2r^2theta = 63` --- (2)

Sub (1) into (2)

`theta/2times(32/(theta+2))^2=63`
`(32^2theta)/((2)(theta+2)^2)=63`
`1024theta=126(theta+2)^2`
`1024theta=126theta^2+504theta+504`
`126theta^2+504theta-520=0`
`63theta^2+252theta-260=0`

Using the quadratic formula,

`theta=(-b+-sqrt(b^2-4ac))/(2a)`

`theta=(-252+-sqrt(252^2-4(63)(-260)))/(2times63)`

`theta=(-252+-sqrt129024)/126`

`theta=0.85rad or -4.85rad`

However, since angles must be positive then it can only be `theta=0.85rad`

Therefore, radius is:

`r=32/(0.85+2)`
`r=11.22cm`

Your two possible angles are `0.85rad, 2pi-0.85=5.43rad`