Some red, white, and blue candies were placed in a bowl. Some contain nuts, and some do not. Suppose one of the candies were cho
sen randomly from all the candies in the bowl. According to the table below, if the candy is blue, what is the probability that it does not contain any nuts?
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Since you mentioned calculus, perhaps you're supposed to find the area by integration.
The square is circumscribed by a circle of radius 6, so its diagonal (equal to the diameter) has length 12. The lengths of a square's side and its diagonal occur in a ratio of 1 to sqrt(2), so the square has side length 6sqrt(2). This means its sides occur on the lines and
.
Let be the region bounded by the line
and the circle
(the rightmost blue region). The right side of the circle can be expressed in terms of
as a function of
:
Then the area of this circular segment is
Substitute , so that
Then the area of the entire blue region is 4 times this, a total of .
Alternatively, you can compute the area of in polar coordinates. The line
becomes
, while the circle is given by
. The two curves intersect at
, so that
so again the total area would be .
Or you can omit using calculus altogether and rely on some basic geometric facts. The region is a circular segment subtended by a central angle of
radians. Then its area is
so the total area is, once again, .
An even simpler way is to subtract the area of the square from the area of the circle.