A 0.2 ml dose of a drug is injected into a patient steadily for 0.75 seconds. At the end of this time, the quantity,Q, of the drug in the body starts to decay exponentially at a continuous rate of 0.2 percent per second. Using formulas, express Q as a continuous function of time,t, in seconds.
Q(t)=_______ if 0 is less than t less than 0.75 and Q(t)=________ if 0.75 less than t less than infinity.
To express Q as a continuous function of time, we need to consider two different cases:
Case 1: 0 < t < 0.75 seconds
During this time interval, the drug is being injected steadily, and the quantity Q is increasing. The formula for the drug quantity during this interval is given by:
Q(t) = 0.2 ml + (rate of injection) * t
Since the rate of injection is constant at 0.2 ml/0.75 seconds, we can substitute this value:
Q(t) = 0.2 ml + (0.2 ml/0.75 s) * t
Simplifying further:
Q(t) = 0.2 ml + 0.26667 ml/s * t
Therefore, if 0 < t < 0.75 seconds, Q(t) = 0.2 ml + 0.26667 ml/s * t.
Case 2: t > 0.75 seconds
After 0.75 seconds, the drug starts to decay at a continuous rate of 0.2 percent per second. To express this decay as a continuous function, we can use the mathematical formula for exponential decay:
Q(t) = Q0 * e^(-k * t)
In this formula, Q0 represents the initial quantity at t = 0, and k represents the decay constant. The decay constant can be calculated using the given decay rate of 0.2 percent per second:
k = -0.2% = -0.002
Since the initial quantity Q0 is the value of Q at t = 0.75 seconds, we can substitute these values into the formula:
Q(t) = Q(0.75 s) * e^(-0.002 * (t - 0.75 s))
Now, we need to find Q(0.75 s), which is the quantity of the drug at the end of the injection. Since Q(0.75 s) is the sum of the injected dose and the quantity gained during injection, we can substitute the values:
Q(0.75 s) = 0.2 ml + (0.2 ml/0.75 s) * 0.75 s = 0.2 ml + 0.2 ml = 0.4 ml
Substituting this value into the decay formula:
Q(t) = 0.4 ml * e^(-0.002 * (t - 0.75 s))
Therefore, if t > 0.75 seconds, Q(t) = 0.4 ml * e^(-0.002 * (t - 0.75 s)).
To summarize:
If 0 < t < 0.75 seconds, Q(t) = 0.2 ml + 0.26667 ml/s * t.
If t > 0.75 seconds, Q(t) = 0.4 ml * e^(-0.002 * (t - 0.75 s)).