The time it takes a ski patroller to respond to an accident call has an exponent

Question

The time it takes a ski patroller to respond to an accident call has an exponential distribution with an average equal to 3 minutes. In what time will 84 percent of all ski accident calls be responded to? (Round the intermediate value to 4 decimal places and then round your final answer to 2 decimal places.) If the ski patrol would like to be able to respond to 84 percent of the accident calls within 19 minutes, what does the average response time need to be? (Round the intermediate value to 4 decimal places and then round your final answer to 2 decimal places.)

Explanation / Answer

A)

Note that
          
P(x<x1) = 1 - e^(-x1/u)          

hence,

0.84 = 1 - exp(-x1/3)

Thus,

x1 = -3*ln(0.16) = 5.497744391 = 5.50 minutes [ANSWER]

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B)

Note that
          
P(x<x1) = 1 - e^(-x1/u)          

hence,

0.84 = 1 - exp(-19/u)

Thus,

u = -19/ln(0.16) = 10.36788835 = 10.37 MINUTES [ANSWER]

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