Question

If there is a goat tied to a rectangular barn on a 50 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? If there are regions you can't find the area of, provide as good an estimate as you can. Assume the goat is tied to a corner outside the barn, cannot get in, and that the barn is not grazing area. (Remember, this will be based on parts of circles, no other shapes...the goat's rope will only get shorter when he tries to go around the barn...)
@jhonyy9

Answer 1

7. When the rope goes around the barn, what is the new radius? How much of a circle can it make without hitting the barn or overlapping area you've already found? What is that area?

The new radius is 30. It can go 20 feet without overlapping. The area is (1/4) x 900 x pi = 225pi.

8. When the rope goes around the barn the other way, what is the new radius? How much of a circle can it make without hitting the barn or overlapping area you've already found? What is that area?

it is the same as number seven because it is just a different 4th of the circle. So the radius is still 30. It can still go 20 feet. and the area is (1/4) x 900 x pi= 225




***these answers are correct. my teacher has already reviewed them but you will need to look at them to answer #9***

Answer 2

9. The areas you found in 7 and 8 overlap each other. How much do they overlap? What *approximate* shape do they make? What is that area?

Answer 3

10. What is the total grazing area the goat can reach?

Answer 4

http://www.mathisfunforum.com/showimage.php?pid=288970&filename=barn.gif&preview=true

Heres a picture that will help.

Answer 5

@jhonyy9 @xapproachesinfinity

Answer 6

@Alknowsall

Answer 7

@satellite73

Answer 8

man this is a very com;licated problem lol ;)

Answer 9

This is a pretty tricky problem, but I would set up a coordinate system. There's no set way to do it but one way is to place the origin at the lower right corner where the goat is tied up. So the goat is tied up at (0,0) which is one corner of the barn

Based on the image, the other points are

(0,20) -- top right corner
(-20,20) -- top left corner
(-20,0) -- bottom left corner

Answer 10

after you have a coordinate system set up, you can find the equations of each circle

why do this? because you'll need to figure out where the circles intersect. Using algebra is one way to do that

Answer 11

yeah but dont u have to use some sort of equation idk which one exactly tho :(

Answer 12

the yellow circle (in the image) would be

(x-0)^2 + (y-0)^2 = 50^2
x^2 + y^2 = 2500

since it's centered at (0,0) and has radius of 50

Answer 13

here let me post the other questions to see if the information in them helps you

Answer 14

to find the area of this main circle in the link uv provided you should use this equation ---> A=pie*r2

Answer 15

6. How much of the 50 foot circle can the goat reach without getting interrupted by the barn? What is that area?

3/4x 50^2 xpi= 1875 pi

Answer 16

the green and red circles have the same radius (30) just different centers

the red circle is centered at (-20,0) so

(x+20)^2 + y^2 = 900

is the equation of the red circle

Answer 17

x^2 + (y-20)^2 = 900 is the equation of the green circle

Answer 18

3/4x 50^2 xpi= 1875 pi for this equation i think x=7.39 but im not sure tho

Answer 19

Fairly accurate drawing

Answer 20

I am still very confused

Answer 21

does the last picture @retirEEd posted help at all?

those circles represent where the goat can go (the max distance). Of course, because the barn is in the way, the goat can't go everywhere along each circle. It's more like in this drawing

http://www.mathisfunforum.com/showimage.php?pid=288970&filename=barn.gif&preview=true

but the equations help you figure out where the circles cross so you can find the approximate area of the overlapping region

Answer 22

@inkyvoyd maybe you can help too?

Answer 23

ok let's go through this one step at a time

here is the barn with corners A,B,C,D

A = (0, 0)
B = (0, 20)
C = (-20, 20)
D = (-20, 0)

see attached

I'm making point A the point where the goat is tied up

Answer 24

if the goat is tied up at point A, and the rope is 50 ft, then it would be able to roam anywhere within this black circle (see attached)

Answer 25

however, the walls of the barn get in the way, so in reality, the goat can trace out this 3/4 circle in red

Answer 26

when the goat reaches the positive y axis, the rope will start to wrap around point B. If the goat keeps going to the left of the positive y axis, then you get this quarter circle in green

Answer 27

repeat these steps but now go clockwise (instead of counterclockwise) and you get this quarter circle in blue

Answer 28

@Anaise how so?

Answer 29

also is there like a equation to set up that I can you to see this in numbers?

Answer 30

I am just all around confused by this. I know it should be simple but I feel like I'm making it harder than it is

Answer 31

the area you want to find out is the area of the two overlapping regions (formed by the blue and green quarter circles)

the yellow region shown

Answer 32

notice how the corner points of this yellow region are C, E, F, G

The coordinates of the points are

C = (-20, 20)
E = (-30, 20)
F = (-20, 30)
G = (-28.71, 28.71)

point G is found by solving the system of equations
(x+20)^2 + y^2 = 900
x^2 + (y-20)^2 = 900

Answer 33

now in #9, they're asking for the approximate area

so it sounds like we can sorta take a shortcut and form the kite CEGF, then use the area of a kite formula to find the approximate area. This isn't perfect of course, but it's an approximation so it should work on some level

Answer 34

You can ignore the circle only advice and draw the radii from DG and BG and have triangles to find the areas of to get an exact result.

Answer 35

It's more work though... aprox. is close and prob. better.

Answer 36

Katie: Next time, PLEASE sketch (or present) the situation described in the problem right up front. I'd bet much info about this would become apparent from a sketch. It seems as though you've left all the sketching to other people to do.

What are the limitations on the areas which the goat can reach? That's a relatively easy question, given that the goat is tied up with a 50-foot rope.

Answer 37

so did you guys got the problem solved?

Answer 38

I can see how to, but after the question closed gave up.

It is essentially three areas added...
Area1 = (3/4)(pi)(50^2) three quarters of the large radius
Area2 = (1/4)(pi)(30^2) one quarter of the a medium radius circle
Area3 = (1/4)(pi)(30^2) one quarter of the other medium radius circle

together minus an OVERLAP half circle
Area4 = (1/2)(pi)(10^2) one half of the a medium radius circle

minus approximately half a square where the two medium radius circles OVERLAP

If you look up the post to my fairly accurate drawing attachment you will the last overlap.
I am not smart enough to calculate that area, but it looks triangular.