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A cow is tethered to one corner of a square field with side 20m with a rope equal to the length of a side of the field. Another cow is tethered at the diagonally opposite corner with the grazing area just touching each other. What is the total grazing area?
- a)200π[2 − √2]
- b)200π[4 − √2]
- c)100π[2 − √2]
- d)150π[2 − √2]
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A cow is tethered to one corner of a square field with side 20m with ...
To solve this problem, we need to find the total grazing area of the two cows. Let's break down the solution step by step:
Step 1: Understanding the problem
We have a square field with side length 20m. One cow is tethered to one corner of the field with a rope equal to the length of a side of the field. Another cow is tethered at the diagonally opposite corner, with the grazing area just touching each other. We need to find the total grazing area.
Step 2: Visualizing the situation
Let's draw a diagram to better understand the situation:
```
+---20m---+
| |
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20m 20m
| |
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+---20m---+
```
Here, the square field has side length 20m, and each side is labeled as 20m. One cow is tethered to one corner (let's call it A) and the other cow is tethered to the diagonally opposite corner (let's call it B). The grazing area is where the two cows can reach while staying within their tethered distance.
Step 3: Calculating the grazing area
To calculate the grazing area, we need to determine the shape formed by the overlapping region of the two tethered circles.
Step 3.1: Finding the radius of the tethered circle
The rope length is equal to the side length of the square field, which is 20m. This means the radius of the tethered circle for each cow is also 20m.
Step 3.2: Calculating the area of the overlapping region
Since the grazing area is formed by the overlapping region of the two circles, we need to find the area of this region.
The overlapping region can be divided into four parts: two quarter circles and two isosceles right triangles.
Step 3.2.1: Finding the area of the quarter circles
The area of a quarter circle is given by the formula: 1/4 * π * r^2, where r is the radius.
For each cow, the area of the quarter circle can be calculated as: 1/4 * π * (20m)^2 = 100π m^2.
Since there are two cows, the total area of the quarter circles is: 2 * 100π m^2 = 200π m^2.
Step 3.2.2: Finding the area of the isosceles right triangles
The isosceles right triangles are formed by the sides of the square field and the radii of the tethered circles.
The length of the hypotenuse of each triangle is equal to the diameter of the circle, which is 2 * 20m = 40m.
The area of an isosceles right triangle is given by the formula: 1/2 * base * height.
In this case, the base and height of the triangle are both equal to the side length of the square field, which is 20m.
So, the area of each triangle is: 1/2 * 20m * 20m = 200 m^2.
Since there are two triangles, the total area of the triangles is: 2 *
Step 1: Understanding the problem
We have a square field with side length 20m. One cow is tethered to one corner of the field with a rope equal to the length of a side of the field. Another cow is tethered at the diagonally opposite corner, with the grazing area just touching each other. We need to find the total grazing area.
Step 2: Visualizing the situation
Let's draw a diagram to better understand the situation:
```
+---20m---+
| |
| |
20m 20m
| |
| |
+---20m---+
```
Here, the square field has side length 20m, and each side is labeled as 20m. One cow is tethered to one corner (let's call it A) and the other cow is tethered to the diagonally opposite corner (let's call it B). The grazing area is where the two cows can reach while staying within their tethered distance.
Step 3: Calculating the grazing area
To calculate the grazing area, we need to determine the shape formed by the overlapping region of the two tethered circles.
Step 3.1: Finding the radius of the tethered circle
The rope length is equal to the side length of the square field, which is 20m. This means the radius of the tethered circle for each cow is also 20m.
Step 3.2: Calculating the area of the overlapping region
Since the grazing area is formed by the overlapping region of the two circles, we need to find the area of this region.
The overlapping region can be divided into four parts: two quarter circles and two isosceles right triangles.
Step 3.2.1: Finding the area of the quarter circles
The area of a quarter circle is given by the formula: 1/4 * π * r^2, where r is the radius.
For each cow, the area of the quarter circle can be calculated as: 1/4 * π * (20m)^2 = 100π m^2.
Since there are two cows, the total area of the quarter circles is: 2 * 100π m^2 = 200π m^2.
Step 3.2.2: Finding the area of the isosceles right triangles
The isosceles right triangles are formed by the sides of the square field and the radii of the tethered circles.
The length of the hypotenuse of each triangle is equal to the diameter of the circle, which is 2 * 20m = 40m.
The area of an isosceles right triangle is given by the formula: 1/2 * base * height.
In this case, the base and height of the triangle are both equal to the side length of the square field, which is 20m.
So, the area of each triangle is: 1/2 * 20m * 20m = 200 m^2.
Since there are two triangles, the total area of the triangles is: 2 *
Community Answer
A cow is tethered to one corner of a square field with side 20m with ...
Diagonal = 20√2
length of rope tying the 2nd cow = 20√2 − 20 = 20(√2 − 1)
grazing area of 1st cow = 14π202 = 14π400 = 100π
grazing area of 2nd cow = 14π400(√2−1)2
Total 100π[1 + (√2 − 1)2] = 100π[1 + 2 + 1 − 22]
= 200π[2 − √2]
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A cow is tethered to one corner of a square field with side 20m with a rope equal to the length of a side of the field. Another cow is tethered at the diagonally opposite corner with the grazing area just touching each other. What is the total grazing area?a)200π[2 − √2]b)200π[4 − √2]c)100π[2 − √2]d)150π[2 − √2]Correct answer is option 'A'. Can you explain this answer?
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