Let R denote the circular region bounded by x^2+y^2 = 36. The lines x=4 and y=3 partition R into four regions R1, R2 ,R3 , and R4. Let [Ri] denote the area of region Ri. If [R1]>[R2]>[R3]>[R4] , then compute [R1]-[R2]-[R3]+[R4].
Could someone help me, I don't really get this...
To understand the problem, let's start by visualizing the circular region and the lines that partition it.
The equation x^2 + y^2 = 36 represents a circle with center (0, 0) and radius 6. This circular region is bounded by this equation.
Now let's look at the lines x = 4 and y = 3. The line x = 4 is a vertical line passing through the point (4, 0), and the line y = 3 is a horizontal line passing through the point (0, 3).
These two lines divide the circular region into four parts: R1, R2, R3, and R4.
To solve the problem, we need to determine the areas of these four regions and calculate [R1] - [R2] - [R3] + [R4].
1. R1 (top-right region):
R1 is the region above the line y = 3 and to the right of the line x = 4. It is a quarter of the circle with radius 6, so its area is (1/4) * π * 6^2.
2. R2 (top-left region):
R2 is the region above the line y = 3 and to the left of the line x = 4. It is a segment of the circle with radius 6, cut by the vertical line x = 4. To find its area, we need to calculate the area of the sector and subtract the area of the triangle formed by the radius and the chord. The angle of the sector can be found using trigonometry as arccos(4/6). So the area of R2 is (arccos(4/6) / (2π)) * π * 6^2 - 1/2 * 2 * 4 * (6 - 3).
3. R3 (bottom-left region):
R3 is the region below the line y = 3 and to the left of the line x = 4. It is a segment of the circle with radius 6, cut by the horizontal line y = 3. Similar to R2, we can calculate its area using the same procedure: (2π - arccos(4/6)) / (2π) * π * 6^2 - 1/2 * 2 * 3 * (6 - 4).
4. R4 (bottom-right region):
R4 is the region below the line y = 3 and to the right of the line x = 4. It is a quarter of the circle with radius 6, similar to R1: (1/4) * π * 6^2.
Now we can calculate [R1] - [R2] - [R3] + [R4] by substituting the areas we found into the expression: [R1] - [R2] - [R3] + [R4] = (1/4) * π * 6^2 - [(arccos(4/6) / (2π)) * π * 6^2 - 1/2 * 2 * 4 * (6 - 3)] - [(2π - arccos(4/6)) / (2π) * π * 6^2 - 1/2 * 2 * 3 * (6 - 4)] + (1/4) * π * 6^2.
By simplifying this expression, you can find the value of [R1] - [R2] - [R3] + [R4].
Sure, I can help you with that!
To find the areas of the four regions, R1, R2, R3, and R4, we need to visualize the circular region bounded by x²+y²=36 and the lines x=4 and y=3.
First, let's draw a rough sketch of the circular region and the lines:
```
---------------------------------
| R2 |
|-------------------------------|
| R3 |
|-------------------------------|
| | | |
| R4 | R1 |
| | | |
|-------------------------------|
```
We can see that R2 lies to the left of the line x=4, R3 lies below the line y=3, R1 lies to the right of the line x=4, and R4 lies above the line y=3.
To find the areas of these regions, we can use integration. However, in this case, it's easier to visualize the regions as geometric shapes and calculate their areas directly.
R2 is the left half of the circular region. Since the circle has a radius of 6 (since x²+y²=36), R2 will have a width of 4 (from x=0 to x=4) and a height equal to the diameter of the circle.
Therefore, the area of R2 is (1/2) * (4) * (12) = 24.
Similarly, R3 is the lower half of the circular region. It will have a width equal to the diameter of the circle (12) and a height of 3 (from y=0 to y=3).
The area of R3 is (1/2) * (12) * (3) = 18.
R1 is the right half of the circular region. Its width is (12 - 4) = 8 (from x=4 to x=12) and has the same height as R2.
The area of R1 is (1/2) * (8) * (12) = 48.
Lastly, R4 is the upper half of the circular region. It has a width equal to the diameter of the circle (12) and a height of (12 - 3) = 9 (from y=3 to y=12).
The area of R4 is (1/2) * (12) * (9) = 54.
Now that we have the areas of R1, R2, R3, and R4, we can calculate [R1] - [R2] - [R3] + [R4]:
[R1] - [R2] - [R3] + [R4] = 48 - 24 - 18 + 54 = 60.
Therefore, [R1] - [R2] - [R3] + [R4] is equal to 60.
I hope this helps clarify the concept for you! Let me know if you have any further questions.
Not sure how to do this without calculus; can't find any theorems about areas cut by intersecting chords.
So, consider the case where the lines x=h and y=k form the chords.
However, with calculus, you can "easily" show that R1+R4 = 1/2 pi r^2 + 2kh
That means that the rest of the circle, R2+R3 = 1/2 pi r^2 - 2kh
So, R1+R4 - (R2+R3) = 4kh
That is, the difference is just that of a rectangle 2h by 2k.