Maximum area of a fenced-in half a circle

In summary: You are allowed to ignore the 2*radius in the expression for the perimeter of the semi-circle because the fence only needs to surround the semi-circle and not the entire enclosed area. Therefore, in summary, you can ignore the 2*radius in the expression for the perimeter of the semi-circle.
  • #1
Mathman2013
23
1
. Homework Statement

ujdAu5z
Let imagine you have gras field formed as a semi circle and you want to fence in that area.

The fence is connected to a wall, so you only have to fence in the area formed by the semi-circle.

You have to use 60 meters of fence bought at a hardware store.

Homework Equations


Area of semi-circle: A = 1/2*Pi*radius^2

Length of the perimeter of a semi-circle. L = Pi*radius + 2*radius

The Attempt at a Solution



And here is where it become tricky for me.

I know that part of the area which needs to be fenced in is the part formed by the semi-circle and excluding the walled area.

However if the length of the perimeter is L = Pi*radius + 2*radius

Am I allowed to ignore the 2*radius ? In my expression for the perimeter of the semi-circle ? (Because L = Pi*radius + 2*radius how you define length of the perimeter of semi circle according to my textbook. )

Yes or no? Because If I am allowed to ignore it then I get an area of the gras field which is 572 m^2, but if I include the 2*radius in the expression for the Length of the perimeter fence then the area of field is only 212 m^2.

Hope someone here can bring light to my question :)
 
Last edited by a moderator:
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  • #2
Can you post a picture showing the wall and the grass semicircle? If the wall is along the diameter of the semicircle, then you don't need the 2*radius.
 
  • #3
Mathman2013 said:
. Homework Statement

ujdAu5z
Let imagine you have gras field formed as a semi circle and you want to fence in that area.

The fence is connected to a wall, so you only have to fence in the area formed by the semi-circle.

You have to use 60 meters of fence bought at a hardware store.

Homework Equations


Area of semi-circle: A = 1/2*Pi*radius^2

Length of the perimeter of a semi-circle. L = Pi*radius + 2*radius

The Attempt at a Solution



And here is where it become tricky for me.

I know that part of the area which needs to be fenced in is the part formed by the semi-circle and excluding the walled area.

However if the length of the perimeter is L = Pi*radius + 2*radius

Am I allowed to ignore the 2*radius ? In my expression for the perimeter of the semi-circle ? (Because L = Pi*radius + 2*radius how you define length of the perimeter of semi circle according to my textbook. )

Yes or no? Because If I am allowed to ignore it then I get an area of the gras field which is 572 m^2, but if I include the 2*radius in the expression for the Length of the perimeter fence then the area of field is only 212 m^2.

Hope someone here can bring light to my question :)

You do not need anybody here to tell you: the answer is given right in the question itself.
 
Last edited by a moderator:

1. How is the maximum area of a fenced-in half circle calculated?

The maximum area of a fenced-in half circle is calculated by finding the radius of the circle, dividing it by two to get the radius of the half circle, and then using the formula A = πr^2/2 to find the area.

2. Why is the maximum area of a fenced-in half circle important in scientific research?

The maximum area of a fenced-in half circle is important in scientific research as it can be used to optimize space usage and efficiency in experiments or simulations involving circular areas.

3. Can the maximum area of a fenced-in half circle be calculated for any size of circle?

Yes, the maximum area of a fenced-in half circle can be calculated for any size of circle as long as the radius is known.

4. Are there any real-world applications of the maximum area of a fenced-in half circle?

Yes, the concept of the maximum area of a fenced-in half circle can be applied in urban planning to determine the most efficient use of land for circular structures such as parks or roundabouts.

5. How does the maximum area of a fenced-in half circle relate to other geometric concepts?

The maximum area of a fenced-in half circle is related to other geometric concepts such as optimization and the Pythagorean theorem, which can be used to find the radius of a half circle. It is also a special case of the more general concept of maximum area for any given shape.

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