In a square ABCD the shaded region is the intersection of two circular regions centered at A and C. if AB =10 then what is the area of the shaded region ?
A. 25(π−2)
B. 50(π−2)
C. 25π
D. 50π
E. 50
Area of two circular regions centered at A and C = Ar. of sector ABD + Ar. of sector CBD
= \(π * \frac{r^2}{4}\) + \(π * \frac{r^2}{4}\) = \(π * 2 * \frac{100}{4}\)
= 50π
Now,
50π = Non - shaded Ar. of sector ABD + Non - shaded Ar. of sector CBD + 2 * (Area for shaded region)
50π = (Non - shaded Ar. of sector ABD + Non - shaded Ar. of sector CBD + Area for shaded region) + Area for shaded region
50π = Ar. of ABCD + Area for shaded region
50π = 10*10 + Area for shaded region
Area for shaded region = 50π - 100
= 50 (π - 2)
ALTERNATIVELY:
As we can see that area of two circular regions is 50π in which area of shaded region is calculated twice. Thus, area of shaded region has to be subtracted once from 50π.
Looking at answer options except 'B' all are out since only has 50π component in it.
Answer B.
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