Trigonometric (Circular) Functions (1 Viewer)

[jellybelly59]

1. A circular metal plate is cut into two segments along a chord equal in length to the radius. What is the ratio of the areas of the two segments?

2. A sheep grazing in a paddock is tethered to a stake by a rope 20m long. If the stake is 10m from the fence, find the area over which the sheep can graze.

3. Two circles each of radius length 10cm, have their centres 16cm apart. Calculate the area common to each circle.

 

[shady145]

i'd say for the 1st one draw a diagram.
draw a chord, then make a triangle from the centre to both the ends of the chorsd. since they are all equal length it is an equalaterla triangle meaning all angles pi/3. use the formula for the area of a minor segment and the area of a circle and keep going. thats how i'd try first anyway

 

[shady145]

okay. question one
area of minor segment = .5r^2(theta - sin(theta))
area of major segment = area of cirlce - are of minor segment
=pixr^2 - .5r^2(theta - sin(theta))
so ratio of big segment to minor segment is
pixr^2 - .5r^2(theta - sin(theta)) : .5r^2(theta - sin(theta))
cancel the r^2, and sub in pi/3 in as the angle. explained where i got it from in previous post
pi-.5(pi/3-sin(pi/3)) : .5(pi/3-sin(pi/3))
break down into simplest form or put a decimal in. i got 33.68:1

 

[shady145]

question 2 is a lil confusing, is the paddock a perfect square? is the stake in the centre of the field? is the perpendicular distance from the stake to the fence 10metres? i think all these need to be considered before an answer can be achieved

 

[shady145]

to understand this u'll need a diagram, i dnt have a scanner so i cant put mine up.
draw 2 circles overlaping each other.
now the circles meet each other at lets say a and b.
draw a line from the centre to a and b, and draw a line from a to b.
notice there are 2 equal minor sectors formed.
we have the radius now we need the angle aob, o being the centre of any of the circles. (it doesnt matter which one u chose becasue both circles are exactly the same)
okay in ur diagram u should have the total distance from O1 to O2 = 16
work out the perpendicualr distance from O1 or O2 to the chord ab.
u should get 8 as it is the midpt of the 2 circles
now using circle properties which come from 3u, a line from the centre that meets a chord, bisects the chord.
this makes 2 right angled triangles and allows us to find the angle. if u notice the 2 sides we know are 10 and 8, then we know its a pythagarine triangle, sides 6,8,10.
actually u dnt even need that.
u could just use cos(theta)=8/10
theta=0.643501108 radians
thats only half the angle that is producing the minor segments, so multiply by 2
2 theta = 1.287002218 radians
area of minor segment = .5xr^2(theta - sin(theta))
since there are 2 minor segments of equal area then we multiply the above expression by 2, plug the info we know in.
so A = 2x.5x10^2(1.287002218 - sin1.287002218 )
=100(0.327002218)
=32.7002218 cm^2 is the common area.
hope u can understand my explaination =s.

 

[Aquawhite]

Hehe, might I suggest you use LaTeX. ^_^ It's there so you can use symbols and make things a lot easier to understand and see.

 

[shady145]

haha, yea i tried using it a while ago but got stuck so i just left it

 

[Uncle]