cohol 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?
Please do not repost several times
To find the constant k in the equation R = e^(kx), we'll use the information given in part (a), which states that a concentration of 0.03 percent alcohol in the bloodstream results in a relative risk of 1.4.
(a) Let's substitute the values into the equation:
1.4 = e^(k * 0.03)
To solve for k, we need to take the natural logarithm (ln) of both sides of the equation:
ln(1.4) = ln(e^(k * 0.03))
Using the property of logarithms, ln(e^(k * 0.03)) simplifies to k * 0.03:
ln(1.4) = k * 0.03
Now, divide both sides of the equation by 0.03 to solve for k:
k = ln(1.4) / 0.03
Evaluate this expression using a calculator to find the value of k.
(b) To find the concentration of alcohol that corresponds to a relative risk of 100, we can use the same k value from part (a) and the given equation R = e^(kx).
100 = e^(kx)
To solve for x (the concentration of alcohol), we can rearrange the equation:
ln(100) = ln(e^(kx))
ln(100) = kx
x = ln(100) / k
Evaluate this expression using the value of k from part (a) to find x.
(c) The law asserts that anyone with a relative risk of 5 or more should not have driving privileges. We need to find the concentration of alcohol in the bloodstream that corresponds to a relative risk of 5.
Using the equation R = e^(kx), we can set the relative risk (R) to 5 and solve for x:
5 = e^(kx)
Take the natural logarithm of both sides:
ln(5) = ln(e^(kx))
ln(5) = kx
Now, divide both sides of the equation by k:
x = ln(5) / k
Evaluate this expression using the value of k from part (a) to find the concentration of alcohol at which a driver should be arrested and charged with a DUI.