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The area of the circular sector is
From that, the equation for the central angle is
The perimeter of the circular sector is the sum of the arc length and twice the radius.
From that, the equation for the central angle is
The value of r can be found by equating these two expressions for θ.
By any of several means, you can find that r=0.2 or 1.25. The latter value corresponds to the requirement that the central angle be less than 90°. The radius is ...
1.25 miles.
From that, the equation for the central angle is
The perimeter of the circular sector is the sum of the arc length and twice the radius.
From that, the equation for the central angle is
The value of r can be found by equating these two expressions for θ.
By any of several means, you can find that r=0.2 or 1.25. The latter value corresponds to the requirement that the central angle be less than 90°. The radius is ...
1.25 miles.
Answer:
D. 91%
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
In which
P(B|A) is the probability of event B happening, given that A happened.
is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Less than 15 minutes.
Event B: Less than 10 minutes.
We are given the following probability distribution:
Simplifying:
Probability of arriving in less than 15 minutes:
Integral of the distribution from 5 to 15. So
Integral of is
Then
Applying the limits, by the Fundamental Theorem of Calculus:
At ,
At ,
Then
Probability of arriving in less than 15 minutes and less than 10 minutes.
The intersection of these events is less than 10 minutes, so:
We already have the integral, so just apply the limits:
At ,
At ,
Then
If given the train arrived in less than 15 minutes, what is the probability it arrived in less than 10 minutes?
Thus 90.87%, approximately 91%, and the correct answer is given by option D.