Passengers using New York's MetroCard system must swipe the card at a rate between 10 and 40 inches per second, or else the card must be re-swiped through the card reader. Research shows that actual swipe rate by subway riders are uniformly distributed between 5 and 50 inches per second.

A. What is the mean swipe rate?
B. What is the standard deviation of the swipe rate?
C. What are the quartiles?
D. What % of subway riders must re-swipe the cards because they were outside the acceptable range?

To find the answers to these questions, we can use the concept of probability distributions and apply it to the given information.

A. Mean Swipe Rate:
The mean (average) swipe rate can be found by taking the average of the lower and upper limits of the uniformly distributed swipe rate. In this case, the lower limit is 5 inches per second and the upper limit is 50 inches per second.

Mean swipe rate = (lower limit + upper limit) / 2
= (5 + 50) / 2
= 55 / 2
= 27.5 inches per second

Therefore, the mean swipe rate is 27.5 inches per second.

B. Standard Deviation of Swipe Rate:
The standard deviation of a uniform distribution can be calculated using the formula:

Standard Deviation = (upper limit - lower limit) / √12

Standard Deviation = (50 - 5) / √12
= 45 / √12
≈ 12.98 inches per second

Therefore, the standard deviation of the swipe rate is approximately 12.98 inches per second.

C. Quartiles:
To find the quartiles, we can divide the range between the lower and upper limits of the uniform distribution into four equal intervals.

Quartile 1 (Q1) = lower limit + (1/4)*(range)
= 5 + (1/4)*(45)
= 5 + 11.25
= 16.25 inches per second

Quartile 2 (Q2) = lower limit + (2/4)*(range)
= 5 + (2/4)*(45)
= 5 + 22.5
= 27.5 inches per second (which is the mean we calculated earlier)

Quartile 3 (Q3) = lower limit + (3/4)*(range)
= 5 + (3/4)*(45)
= 5 + 33.75
= 38.75 inches per second

Therefore, the quartiles are Q1 = 16.25 inches per second, Q2 = 27.5 inches per second, and Q3 = 38.75 inches per second.

D. Percentage of Subway Riders Outside Acceptable Range:
To find the percentage of subway riders who must re-swipe the cards because they were outside the acceptable range, we need to calculate the probability of the swipe rate being outside the range of 10 to 40 inches per second.

Probability of swipe rate being outside range = (range outside acceptable range) / (total range)

Range outside acceptable range = [5, 10] and [40, 50]
= 5 + (50 - 40) + (10 - 5)
= 5 + 10 + 5
= 20 inches

Total range = 50 - 5
= 45 inches

Probability of swipe rate being outside range = 20 / 45
= 0.4444

To convert this probability to a percentage, we multiply by 100:

Percentage of subway riders outside the acceptable range = 0.4444 * 100
≈ 44.44%

Therefore, approximately 44.44% of subway riders must re-swipe the cards because they were outside the acceptable range.