Respuesta :
Answer:
To find the probability that the seal properly fits the valve, we need to consider the difference between their diameters being within the specified range.
Let's denote the diameter of the valve as \( V \) and the diameter of the seal as \( S \). According to the problem, for the seal to properly fit the valve, the difference between their diameters must be within \( x \) millimeters. Mathematically, this can be expressed as:
\[ |S - V| \leq x \]
Now, since the diameters are chosen at random, we can assume that both the diameter of the valve and the diameter of the seal follow a uniform distribution. Therefore, the probability density function (PDF) for both diameters is constant within their respective ranges.
To find the probability that the seal properly fits the valve, we need to find the proportion of the area where the difference between their diameters is within \( x \) millimeters relative to the total area of the possible combinations of diameters.
This can be visualized as a square with side length \( 2x \), representing the possible range of differences between the diameters. The area of this square represents the total possible combinations of diameters. Within this square, the region where \( |S - V| \leq x \) represents the combinations where the seal properly fits the valve.
Since this region is a smaller square with side length \( x \) inside the larger square, the probability that the seal properly fits the valve is the ratio of the area of the smaller square to the area of the larger square, which is:
\[ \text{Probability} = \frac{x^2}{(2x)^2} = \frac{x^2}{4x^2} = \frac{1}{4} \]
Therefore, the probability that the seal properly fits the valve is \( 0.25 \), which corresponds to option \( \text{a) } 0.25 \).
