Solve the problem

Question

The amount of alcohol in the bloodstream, A, declines at a rate proportional to the amount. That is, dA/dt = kA

If k=-0.3 for a person, how long will it take for his alcohol concentration to decrease from 0.10% to 0.05%?

Answer 1

dA/dt = -0.3A

 

Separate the variables:

 

dA/A = -0.3dt

 

Integrate both sides:

 

ln(A) = -0.3t + ln(C), where C = unknown constant

 

Take e^() on both sides:

 

A(t) = Ce^(-0.3t)

 

Let A(0) = 0.10

 

Let A(T) = 0.05, where T = time when A drops by half (i.e. from 0.10 to 0.05)

 

A(T) / A(0) = 1/2

 

A(T) / A(0) = [Ce^(-0.3T)] / [C] = 1/2

 

e^(-0.3T) = 1/2

 

T = ln(1/2) / (-0.3) = 2.310 time units 

 

No units were given for k, but k must have units of (time units)^(-1), such as sec^-1, min^-1, etc.

 

Note we answered the question without having to find the value of C.