Solve the problem.
-The amount of alcohol in the bloodstream, $\mathrm { A }$, declines at a rate proportional to the amount, that is, $\frac { \mathrm { dA } } { \mathrm { dt } } = - \mathrm { kA }$. If $\mathrm { k } = 0.5$ for a particular person, how long will it take for his alcohol concentration to decrease from $0.10 \%$ to $0.05 \%$ ? Give your answer to the nearest tenth of an hour.
A) 2.8 hr
B) 0.3 hr
C) 2.1 hr
D) 1.4 hr

Question

Quizplus · Accepted Answer

We can use the exponential decay formula for this problem:
$$\mathrm{A(t) = A_0 e^{-kt}},$$
where A(t) is the amount of alcohol in the bloodstream at time t, A0 is the initial amount of alcohol, and k is the rate constant.

We are given that k = 0.5, and we can assume that A0 = 0.10%. Thus, we want to solve for t when A(t) = 0.05%:
$$0.05\% = 0.10\% e^{-0.5t}$$
Simplifying, we get:
$$e^{-0.5t} = 0.5$$
$$t = \frac{-\ln 0.5}{0.5} = 1.386 \approx 1.4\mathrm{~hours}$$
Therefore, the answer is D.

Solve the Problem A\mathrm { A }A , Declines at a Rate Proportional to the Amount, That

Solve the Problem $\mathrm { A }$ , Declines at a Rate Proportional to the Amount, That