lcohol and Driving The concentration of alcohol in a person’s bloodstream is measurable.
Suppose that the relative risk R of having an accident while driving a car can be modeled by the
equation
R = e
kx
where x is the percent of concentration of alcohol in the bloodstream and k is a constant.
(a) Suppose that a concentration of alcohol in the bloodstream of 0.03 percent results in a relative risk
of an accident of 1.4. Find the constant k in the equation.
(b)Using the same value of k, what concentration of alcohol corresponds to a relative risk of 100?
(c) If the law asserts that anyone with a relative risk of having an accident of 5 or more should not
have driving privileges, at what concentration of alcohol in the bloodstream should a driver be
arrested and charged with a DUI?
ignoring e for this, because it seems like well maybe there was a typo?
(a) 0.03% k = 1.4
3/100000 * k = 1.4
k = 46666 2/3
if you meant 3%, then we get 46 2/3 as k
(b)R = 100 = kx
100 = kx
100 = 46 2/3 x => x = 2.142857143....
(or 100= 46666 2/3 x = 0.002142857....)
(c) I guess k would equal 46 2/3?, tell me your thoughts.
Anyways, then 5 = 46 2/3 x,
so x = 0.107142857
Did you mean e as in the mathematical constant? (Euler's number)
Like, the number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm. It is the limit of (1 + 1/n)n as n approaches infinity.
Thanks if you have any additions to make or observations tell me
To find the constant k in the equation R = e^kx, we can use the information given in part (a). We know that a concentration of alcohol in the bloodstream of 0.03 percent results in a relative risk of an accident of 1.4.
Using this information, we can substitute the values into the equation:
1.4 = e^(k * 0.03)
To solve for k, we take the natural logarithm (ln) of both sides of the equation:
ln(1.4) = k * 0.03
Now we can divide both sides by 0.03 to isolate k:
k = ln(1.4) / 0.03
This will give you the value of the constant k.
For part (b), we are given that we need to use the same value of k to find the concentration of alcohol that corresponds to a relative risk of 100. We can use the equation R = e^kx and substitute the relative risk of 100:
100 = e^(kx)
Now, solve for x by taking the natural logarithm of both sides:
ln(100) = kx
Divide both sides by k to isolate x:
x = ln(100) / k
Substituting the value of k that we calculated in part (a), you can now find the concentration of alcohol corresponding to a relative risk of 100.
For part (c), we need to find the concentration of alcohol in the bloodstream at which a driver should be arrested and charged with a DUI. Given that the law asserts a relative risk of 5 or more, we can set up the equation:
5 = e^(kx)
To solve for x, take the natural logarithm of both sides:
ln(5) = kx
Divide both sides by k to isolate x:
x = ln(5) / k
Using the value of k that we calculated in part (a), this will give us the concentration of alcohol in the bloodstream at which a driver should be arrested and charged with a DUI.