To represent the quantity $Q$ of a drug in the bloodstream as a function of time $t$ using the form $Q = ab^{-t/T}$, we need to determine the constants $a$, $b$, and $T$ given the initial conditions and the decay behavior of the drug.

Given Information:

• Initial quantity of the drug: $Q(0) = 300$ mg.
• The drug decays to one-fifth of its value every 120 minutes.

Function Form:

The function form given is:

$Q(t) = ab^{-t/T}$

where:

• $a$ is the initial quantity of the drug.
• $b$ is the base of the exponential decay.
• $T$ is the time constant associated with the decay.

Step-by-Step Solution:

1. Determine $a$:

   The initial quantity of the drug in the bloodstream is 300 mg. Therefore, when $t = 0$:

   $Q(0) = ab^{0} = a \times 1 = a$

   So, $a = 300$.

2. Determine $b$ and $T$:

   The problem states that the drug decays to one-fifth of its value every 120 minutes. This means:

   $Q(120) = \frac{1}{5} Q(0)$

   Substituting $Q(0) = 300$ and the functional form:

   $Q(120) = 300b^{-120/T} = \frac{1}{5} \times 300$

   Simplifying this:

   $300b^{-120/T} = 60$

   Dividing both sides by 300:

   $b^{-120/T} = \frac{60}{300} = \frac{1}{5}$

   Taking the reciprocal to simplify:

   $b^{120/T} = 5$

3. Solve for $b$:

   We know that $b^{120/T} = 5$, and since we want a base for our exponential decay:

   $b = 5^{T/120}$

   To maintain a consistent decay form, we can choose $T = 120$ because every 120 minutes, the drug reduces to one-fifth:

   $b = 5^{120/120} = 5$

4. Final Values for $a$, $b$, and $T$:

   • $a = 300$ (initial quantity of the drug)
   • $b = 5$ (base of the exponential decay)
   • $T = 120$ (time constant for decay to one-fifth)

Final Function:

The function that represents the quantity $Q$ of the drug in the bloodstream over time $t$ is:

$Q(t) = 300 \times 5^{-t/120}$

Or, equivalently:

$Q(t) = 300 \times \left(\frac{1}{5}\right)^{t/120}$

Would you like further details, or do you have any questions?

Related Questions:

1. How does changing the decay time constant $T$ affect the rate at which the drug decays?
2. What would the function look like if the drug decayed to one-tenth of its value every 120 minutes?
3. How can we convert this exponential decay model into a continuous decay model using $e$?
4. What is the half-life of the drug according to this model?
5. How much of the drug remains in the bloodstream after 240 minutes?

Tip: Understanding exponential decay is crucial in pharmacokinetics, where it models how drugs dissipate in the body.