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Let R be the circle centered at (0,0) with radius  10. The lines x=6 and y=4 divide R into four regions R1, R2, R3 , and R4. Let  R_i denote the area of region R_i If R1>R2>R3>R4,
then find R1-R2-R3+R4

 Oct 28, 2023
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The lines x=6 and y=4 divide the circle R into four regions, as shown below:

We can see that R1​ is the largest region, followed by R2​, R3​, and R4​.

To find the areas of these regions, we can use the following formula for the area of a sector of a circle:

Area of sector = (Central angle / 360 degrees) * Area of circle

The central angles of the four sectors are as follows:

R1​: 90∘

R2​: 45∘

R3​: 45∘

R4​: 90∘

The area of the circle is πr2=π⋅102=100π.

Therefore, the areas of the four regions are as follows:

R1​=(90∘/360∘)⋅100π=25π

R2​=(45∘/360∘)⋅100π=12.5π

R3​=(45∘/360∘)⋅100π=12.5π

R4​=(90∘/360∘)⋅100π=25π

Therefore, R1​>R2​>R3​>R4​, and R1​−R2​−R3​+R4​=25π−12.5π−12.5π+25π=25π​.

 Oct 28, 2023

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