Alonzo Odom

2022-07-20

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 an equation of the form $R={e}^{kx}$ where x is the percent of concentration of alcohol in the bloodstream andk is a constant. 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.

Kyan Hamilton

To solve the problem, substitute the given values of x and R in the given equation and solve for the value of k.
$1.4={e}^{0.03k}$ Rewrite in natural logarithmic form.
$0.03k=\mathrm{ln}1.4$ Divide both sides by 0.03
$\frac{0.03k}{0.03}=\frac{\mathrm{ln}1.4}{0.03}$
$k=\frac{0.3365}{0.03}$
k=11.22
Thus, the value of the constant k is 11.22
Result:
11.22

Substitute 0.03 to x and 1.4 to R in the given equation to find the value of k:
$R\right)={e}^{kx}$
$1.4={e}^{k\left(0.03\right)}$
$1.4={e}^{0.03k}$
Use the rule $a={e}^{b}⇒\mathrm{ln}a=b$ to obtain:
$\mathrm{ln}1.4=0.03k$
$\frac{\mathrm{ln}1.4}{0.03}=\frac{0.03k}{0.03}$
$\frac{\mathrm{ln}1.4}{0.03}=k$
Use a calculator to obtain:
k=11.21574122
$\approx 11.2157$
Thus, $k=\frac{\mathrm{ln}1.4}{0.03}\approx 11.2157$
Result:
$k=\frac{\mathrm{ln}1.4}{0.03}\approx 11.2157$

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