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The problem asks us to express the quantity of a drug, \( Q(t) \), in the body as a function of time, \( t \), with two different expressions: one for the interval when the drug is being injected and another for when the drug starts to decay exponentially.

### Given Information:
- A **0.4 ml dose** of the drug is injected steadily over **0.6 seconds**.
- After this, the drug decays **exponentially** at a rate of **0.3% per second** (or \( 0.003 \) as a decay constant).
- We are asked to provide \( Q(t) \) for two cases:
  - \( 0 \leq t \leq 0.6 \) (during the injection).
  - \( 0.6 < t \leq \infty \) (exponential decay after injection).

### Solution Breakdown:
1. **During the injection ( \( 0 \leq t \leq 0.6 \) ):**
   - The drug is injected steadily over 0.6 seconds.
   - Therefore, the amount of the drug \( Q(t) \) increases linearly with time during this interval.
   - The function can be expressed as:
     \[
     Q(t) = \frac{0.4}{0.6} t = \frac{2}{3} t
     \]
   
2. **After the injection ( \( 0.6 < t \leq \infty \) ):**
   - After the drug injection stops, it decays exponentially with a rate of 0.3% per second.
   - The initial quantity at \( t = 0.6 \) is \( Q(0.6) = 0.4 \) ml.
   - The decay can be modeled using the exponential decay formula:
     \[
     Q(t) = 0.4 e^{-0.003(t - 0.6)}
     \]
   - Here, \( t - 0.6 \) accounts for the time starting from 0.6 seconds, and \( e^{-0.003} \) represents the decay at 0.3% per second.

### Final Functions:
- For \( 0 \leq t \leq 0.6 \):
  \[
  Q(t) = \frac{2}{3} t
  \]
- For \( 0.6 < t \leq \infty \):
  \[
  Q(t) = 0.4 e^{-0.003(t - 0.6)}
  \]

These are the expressions needed to describe \( Q(t) \) in both intervals.

Would you like any more details on this problem? Here are some related questions you might find helpful:

1. How is exponential decay applied in real-world scenarios?
2. What is the significance of the decay constant in an exponential model?
3. How can linear and exponential functions be compared in different contexts?
4. How does continuous drug infusion compare to a single-dose model?
5. How does the unit of time affect the exponential decay formula?

**Tip:** In exponential decay, the decay constant \( k \) determines how quickly the substance decreases over time. The smaller the constant, the slower the decay.

A 0.4 ml dose of a drug is injected into a patient steadily for 0.6 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 0.3 percent per second. Using formulas, express Q as a continuous function of time, t, in seconds.

Explore how to model the concentration of a drug in the body over time using linear and exponential functions. This problem provides formulas for the steady injection and subsequent exponential decay of a drug, suitable for college-level studies in calculus and differential equations.

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