Need help with trig

[HERculina]

Hey guys,
Can you please help me figure out how to work out this maths question in relation to arcs, sectors and segments:
Two irrigation sprinklers spread water in circular paths with radii of 7m and 4m. If the sprinklers are 10m apart, find the area of crop that recieves water from both sprinklers.



Answer: 2.95 m^2

[david10d]

Draw a diagram and find the area of the intersection of the two circles. Cbf working it out, but that's the method that should be used.

[brightsky]

Hey guys,
Can you please help me figure out how to work out this maths question in relation to arcs, sectors and segments:
Two irrigation sprinklers spread water in circular paths with radii of 7m and 4m. If the sprinklers are 10m apart, find the area of crop that recieves water from both sprinklers.



Answer: 2.95 m^2

Joint radius = 7 + 4 = 11. However the radii are 10 m apart. Hence there is 1 m overlap.

Let the two points where the circles overlap be A and B from top to bottom and the centre of the smaller circle be O_1 and centre of larger be O_2.

Draw O_1A , O_1B , O_2A and O_2B. Also draw AB.

Find AB.

Then use that and the cosine rule to find angle AO_1 B and AO_2B.

Then the area would be:

Sector (AO_2B) + Sector (AO_1B) - Area of kite O_1AO_2B

[HERculina]

Wow thanks! I would have never thought of joining the centre of the two circles with the top and bottom of the intersecting middle bit to form different sectors.

Draw O_1A , O_1B , O_2A and O_2B. Also draw AB.

Find AB.

But how do I find AB? Can you give me a clue on wat rule or method Im suppose to use?
This is so hard 

[brightsky]

Wow thanks! I would have never thought of joining the centre of the two circles with the top and bottom of the intersecting middle bit to form different sectors.

Draw O_1A , O_1B , O_2A and O_2B. Also draw AB.

Find AB.

But how do I find AB? Can you give me a clue on wat rule or method Im suppose to use?
This is so hard 

Pythag would do I think.

[pi]

(assuming you drew what brightsky said)

Take the 7cm radius circle.
Use pythagoras with sides 7cm (as the hypotenuse), 6.5cm (7cm - half the length of overlap) and x (as half of AB).

Solve for x and multiply by 2 for AB

EDIT: completely wrong, only works for circles with same radii

[HERculina]

Ohhh, but how do you know that you have to minus half the length of the overlap. How do you know AB cuts through the centre of the 1m line?
Sorry, I take long to process things  

Is the diagram meant to be like a venn diagram. Similar to the one below?

[pi]

You are right, its not 0.5. I should have drawn a diagram...  

Using your diagram:

Get the area of the triangle (half of kite cut through the dotted line O₁/O₂) though Heron's formula:
where and .

Multiply by 2 to get area of kite.

I think that will work rather than finding AB.

EDIT: Thought other radii was 11, when it was 4

Then continue with brightsky's methods:

Then the area would be:

Sector (AO_2B) + Sector (AO_1B) - Area of kite O_1AO_2B

[HERculina]

I was wondering how to work out the area of the kite. xD
At first, I thought of it as a trapezium looking diagonally :idiot2:
then realized it was too difficult; i was going to use the other 2 triangles with the AB lines in each of them to work out the area of the kite.
BUT dont we still need to know what AB is in order to use the cosine rule and work out the angles O1 and O2 and therefore the areas of both sectors using the sector rule Area=1/2r^2θ?  :buck2:

[Andiio]

Ohhh, but how do you know that you have to minus half the length of the overlap. How do you know AB cuts through the centre of the 1m line?
Sorry, I take long to process things 

Is the diagram meant to be like a venn diagram. Similar to the one below?

Yep that's the diagram.

I remember that you can apply the sine/cosine rules etc to solve as well.

[vgardiy]

Simultaneous Equations
y^2 + (10-x)^2 = 49
y^2 + x^2 = 16


solve for y (and x maybe)

and AB=2y

[HERculina]

(Image removed from quote.)

Simultaneous Equations
y^2 + (10-x)^2 = 49
y^2 + x^2 = 16


solve for y (and x maybe)

and AB=2y

wat a genius!  

[pi]

^^ go here for the general solutions if you like that way (its a neat way to go about it)

[HERculina]

^^ go here for the general solutions if you like that way (its a neat way to go about it)

that looks a tad difficult...

[HERculina]

Hey again, can you help me out with another trig question:

Ive worked out a) and  b ) out already btw where i got a) 3.931 km  by using the sine rule and for b) 6.075 km^2 by using the formula
Area = 1/2 absinC. So i just need help for q.s c) and d)

A farmer owns a large triangular area of flat land, bounded on one side by an embankment to a river flowing NE, on a second side by a road which meets the river at a bridge where the angle between river and road is 105 degrees, and on the third side by a long fence. Find:

a) the length of the river frontage, correct to 3 decimal places
b) the area of the land correct to 3 decimal places

The farmer decides to divide the land into two sections of equal area, by running a fence from the bridge to a point on the opposite side.
c) What bearing must the fence be built?
d) What is the length of the fence, correct to 3 decimal places?

Answers: c)North 89 degrees 53 minutes East     d) 2.190 km

THANKS

See diagram below: