Answer:
p(2 in shaded) = 9x²/(144x² +96x +16)
Explanation:
Given a rectangle of dimensions (6x+2) by (2x+2) containing a shaded rectangle of dimensions (3x) by (x+1), you want the probability that two randomly placed darts will fall within the shaded area.
Shaded area
The fraction of the total area that is shaded is ...
shaded area / total area = (3x)(x+1)/((6x+2)(2x+2)) = (x+1)(3x)/((x+1)2(6x+2))
= 3x/(12x+4) . . . . . factors of x+1 cancel
Probability
The probability a randomly placed dart will be placed in the shaded area is equal to the fraction of the area that is shaded. The probability that two darts will land there is the product of the probabilities:
p(2 in shaded) = p(1 in shaded) × p(1 in shaded) = p(1 in shaded)²
In terms of x, this is ...
p(2 in shaded) = (3x)²/(12x +4)² = 9x²/(144x² +96x +16)
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