I couldn't do my math homework. Can you help me please?

Question

A cow is tethered to a post alongside a barn 10 m wide and 30 m long. If the post is 10 m from a corner of the barn and if the rope is 30 m long, find the cow's total grazing area.

Answer 1

The area of a circle is pi*radius^2 (^ is power operator ^2 is squared). The radius is the distance from the center of the circle to one point on the circle.

If you look at the picture below you can see the area where the cow can graze are divided into 4 areas. Area A is the front of barn where the cow can use move his rope freely. This is a half circle. The areas B and C are the area where his rope is up against the barn on one side allowing the cow to move in a quater circle until the rope is against two side of the barn. Area D is where on one side the cow has remaining rop to graze in another quater cirlce util the rope is up against three sides of the barn.

So the total grazing area is
TotalArea = A + B + C + D

A = pi * (30^2) / 2 = 1413.72 (half of the area of a circle)
B = pi * (20^2) / 4 = 314.16 (quarter of the area of a cicle)
C = pi * (10^2) /4 = 78.54
D = pi * (10^2) / 4 = 78.54

TotalArea = 1884.96


The ascii art below needs a fixed width font not sure how to do that in answers so you can cut and paste it into notepad to view it.

----------------------------------rope
-------------------------#.....#...30....#...#
--------------------#..............|..............#
---------------#.......Area.A......|...................#
-----------#.......................|.......................#
--------#..........................|.........................#
-----#.............................|............................#
---#...............................|..............................#
--#................................|...............................#
-#.............<===20=============>|<=====10====>...................#
<==10=========>___________________0____________<----20------------->
-#........./\.|::::::::::::::::::::/\::::::::::|....................#
-#..Area.C.|..|::::::::::::::::::::|:::::::::::|......Area.B........#
--#........10.|::::::::::Barn::::::|:::::::::::|....................#
---#.......|..|::::::::::::::::::::10::::::::::|...................#
-----#.....|..|::::::::::::::::::::|:::::::::::|...................#
-------#.#.\/.|____________________\/__________|..................#
------------------------------------<===10======>/\...............#
------------------------------------#.....Area.D.|...............#
--------------------------------------#..........10............#
-----------------------------------------#.......|..........#
--------------------------------------------#..#.\/#....#

-rope
..30....#...#
..|..............#
..|...................#
..|.......................#
..|.........................#
..|............................#
..|..............................#
..|...............................#
=>|<=====10====>...................#
_0____________<----20------------->
::/\::::::::::|....................#
::|:::::::::::|......Area.B........#
::|:::::::::::|....................#
::10::::::::::|...................#
::|:::::::::::|...................#
__\/__________|..................#
---<===10======>/\...............#
---#.....Area.D.|...............#
-----#..........10............#
--------#.......|..........#
-----------#..#.\/#....#

Answer 2

First draw a rectangle 10 metres by 30 metres to represent the barn. Then put a dot on the broadside of the barn to represent the tethering post. It is 10m from 1 end, 20m from the other.
Then draw a dashed line along the side of the barn to represent the cow's rope. It should go along the 20m and 10m further. Then draw a circle with the post as the centre and the radius reaches to the point 10m past the barn. The circle also reaches 20m past the opposite end of the barn and completely surrounds the barn.
If there was no barn, the cow could graze the full circle. With the barn there, he can only travel around the barn as far as the 30m rope will wrap around.
The area is found by sectioning the accessible area into half circles and quarter circles. She can obviously graze half of the big circle that is marked by drawing a line along the broad side of the barn. This half circle has area one half Pi radius squared {(1/2)*Pi*r^2} = (1/2)*Pi*(30)^2 = (1/2)*Pi*900 = 450*pi square metres.
Then we need to section off 3 quarter circles to find the areas the cow can reach while wrapping the rope around the barn. first if the cow walks the 20m along the broad side, she can reach 10m past the end. She can then walk 10 more mteres in the same direction, or walk along the 10m short side of the barn, or anywhere in the quarter circle thus defined. This area is (1/4)*Pi*(10)^2 = (1/4)*Pi*100 = 25*Pi square metres.
The cow can travel further around the other end of the barn (where the post is 10m from the end. After walking 10m to the end of the barn, the cow can walk 20m more in the same direction or walk 20m along the short side of the barn. This defines a quarter circle of 20m radius. The area is (1/4)*pi*(20)^2 = (1/4)*Pi*(400) = 100*Pi square metres.
There's one more quarter circle because the cow could walk 20m along the short side and therefore 10m past the short side. She can then wrap around 10m along the broad side opposite the post and can reach another quarter circle of radius 10m. As before this is an area of 25*Pi square metres.
Adding all the areas we get: 450*Pi + 25*Pi + 100*Pi + 25*Pi
=600*Pi square metres.
This is approx. 1884 square metres.

Answer 3

kind of a trick question.

Assume post is "alongside" meaning against the side of the barn.

Assume it is along the "long" wall since the short wall is only 10m and the post would be "at the other corner" if 10m down the short wall.

Now when you draw a picture, notice the cow can go around the ends of the barn wrapping the rope around as it goes. There are areas the cow can reach on all four sides of the barn and each needs to be calculated (using pi*r^2).

I get about 1885 meters squared

Answer 4

This problem has multiple solutions depending upon where the post is located. For example, 10 m from a corner of the barn could be 10 down the side of the barn (which is still 10 m from the corner), this will yield a different grazing area than if the post is out from the side of the barn some distance which is not specified in this question.

Answer 5

If the area of a circle is pi x R^2 and R=30 the entire circle's area is 900 x PI squared Meters less (20 x cos 45) for the area of the circle is taken up by the building that the cow cannot graze.

Answer 6

The position of the post is not clear. Is the post in line with one of the sides of the barn or does it go out at an angle to both sides of the barn?
Putting it simply, it the radius of a 30m circle minus the over lap area of the barn.

Answer 7

well obviously the post is on the 30m side of the barn 10m from a corner. the cow would then have 1/2 of a 30m radius circle and 1/4 of a 20m radius circle , 1/4 of a 10m radius circle (on the short side) and 1/4 of a 10m radius circle (on the other side) (envision it). u do the rest

Answer 8

If you plot it out to scale on paper (say 1 " = 10m)
it will be easy to find the answer - you will see it.

Remember the formula for the area of a crcle is 3.14 x radius squared.

;-)

Answer 9

rectangular area=10*30 =300
circle area=pi*30*30=900*3.14 = 826

the the rest by yourself.

Answer 10

Depends if there is any grass around the barn. Cannot answer without more information.