# The Exponential Function and Its Inverse

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• Dec 2nd 2009, 09:33 PM
darksoulzero
The Exponential Function and Its Inverse
Hi I got 3 questions that I am stuck on in my homework. Any help explaining how you came across the answer would help. Thanks in advance.

1. The probability, P percent, of having an accident while driving a a car is related to the alcohol level of the driver's blood by the formula P= e^(kt), where k is the constant. Accident statistics show that the probability of an accident is 25% when the blood alcohol level is t = 0.15.

a) Find k. Use P=25
b) At what blood alcohol level is the probability of having an accident 50 %.

2. A failure of o-ring seals caused the challenger space shuttle disaster in 1986. After the disaster, there was a study of 23 shuttle launches that had preceded the fatal flight. A mathematical model was developed involving the relationship between the Celsius temperature, x, around the o-rings and the number, y, of eroded of leaky primary o-rings. The model sated that y = ( 6) / (1 + e^[-{5.085-0.1146x}]), where the number 6 indicates the 6 primary o-rings on the spacecraft.

a) What is the predicted number of eroded or leaky primary o-rings at a temperature of 37.5 degrees Celsius.
• Dec 2nd 2009, 10:15 PM
earboth
Quote:

Originally Posted by darksoulzero
Hi I got 3 questions that I am stuck on in my homework. Any help explaining how you came across the answer would help. Thanks in advance.

1. The probability, P percent, of having an accident while driving a a car is related to the alcohol level of the driver's blood by the formula P= e^(kt), where k is the constant. Accident statistics show that the probability of an accident is 25% when the blood alcohol level is t = 0.15.

a) Find k. Use P=25
b) At what blood alcohol level is the probability of having an accident 50 %.

2. A failure of o-ring seals caused the challenger space shuttle disaster in 1986. After the disaster, there was a study of 23 shuttle launches that had preceded the fatal flight. A mathematical model was developed involving the relationship between the Celsius temperature, x, around the o-rings and the number, y, of eroded of leaky primary o-rings. The model sated that y = ( 6) / (1 + e^[-{5.085-0.1146x}]), where the number 6 indicates the 6 primary o-rings on the spacecraft.

a) What is the predicted number of eroded or leaky primary o-rings at a temperature of 37.5 degrees Celsius.

to #1. a):

Plug in the given values and solve for k:

$25 = e^{k \cdot 0.15}~\implies~\ln(25)=0.15 \cdot k$

I'll leave the rest for you.

b) Plug in P = 50 and the k-yalue from 1.a) and solve for t:

$50 = e^{\frac{20}3 \cdot \ln(25) \cdot t}~\implies~\ln(50)=\frac{20}3 \cdot \ln(25) \cdot t$

Solve for t.

to #2: You only have to plug in all given values. Use a calculator!
• Dec 3rd 2009, 10:26 AM
darksoulzero
Thanks For the help. I found out that e was a constant like pi; I found out how to do it. However, for question 1, I can not use ln because it wasn't taught.