Does Rover have enough roaming room? How much roaming room does Rover have? You must answer both questions. Write an explanation of how you got your answer. (Details Below)

This is a regular hexagon, so all sides are of equal length.

The perimeter of the hexagon is 78 metres, and it has 6 sides.

Each side is:

#78/6=13 " metres"#

The sum of the interior angles of a regular polygon is given by:

#108^@n-360^@#

Where #bbn# is the number of sides.

Sum of angles:

#180^@*6-360^@=720^@#

Each angle:

#(720^@)/6=120^@#

The length of chain is 12 metres.

From diagram:

#AC=AB+AD=12#150.7964474" units"^2

Since #AC< AE#

The chain describes an arc:

#hat(BCD)#

#/_BAD# is the angle subtended by this arc.

#/_BAD=120^@#

The area that Rover has to roam is the area of the sector:

#Ahat(BCD)#

We can find this area by using:

Area of a sector is:

#1/2r^2theta#

Where #theta# is the angle subtended by the arc measured in radians.

Convert #120^@# to radians:

#(120^@*pi)/(180^@)=(2pi)/3#

Our radius is #AC=12#

Area of sector:

#"Area"=1/2(AC)^2((2pi)/3)=1/2*144*(2pi)/3=48pi" units"^2#

This is approximately:

#48pi~~150.7964474" units"^2#

This is greater than the recommended area of #150m^2#, so the answer is yes, Rover does have enough room to roam.