A 0.3 ml dose of a drug is injected into a patient steadily for 0.35 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 0.4 percent per second. Using formulas, express Q as a continuous function of time, t, in seconds.

Hi Brittany, you're going to need a split function, one where t <= 0.35 and one where t > 0.35.

a) When t <= 0.35
The information you have is: when t = 0, the quantity will also be zero. When t = 0.35 the quantity will be 0.3. Seems like this will be a linear equation

b) When t > 0.35
The information you have that for every second that goes by Q will decrease by .4 percent. So you'll have to take away .4% of Q each t.

Make sense? Let me know if you need more help!

To express the quantity, Q, of the drug in the body as a continuous function of time, t, we need to understand the process of exponential decay.

Exponential decay is a mathematical model that describes the reduction of a quantity over time. In this case, the drug quantity is decaying at a continuous rate of 0.4 percent per second.

The general formula for exponential decay is given by:
Q(t) = Q0 * e^(rt)

Where Q(t) is the quantity at time t, Q0 is the initial quantity at time t=0, e is the base of the natural logarithm (approximately 2.71828), r is the decay constant, and t is the time.

In our case:
- The initial quantity of the drug in the body is 0.3 ml (Q0 = 0.3 ml).
- The decay constant is 0.4 percent per second, which can be converted to its decimal form by dividing by 100 (r = 0.4/100 = 0.004).

Therefore, the expression for Q(t) becomes:
Q(t) = 0.3 * e^(0.004t)

So, the continuous function expressing Q as a function of time, t, is Q(t) = 0.3 * e^(0.004t), where t is measured in seconds.

To express the quantity of the drug, Q, as a continuous function of time, t, in seconds, we can use the formula for exponential decay:

Q(t) = Q₀ * e^(kt)

Where:
- Q(t) is the quantity of the drug in the body at time t.
- Q₀ is the initial quantity of the drug.
- e is the base of the natural logarithm, approximately 2.71828.
- k is the decay constant.

In the given scenario, the initial dose of the drug is 0.3 ml, and the decay rate is 0.4 percent per second. To convert the decay rate from percentage to a decimal, we divide it by 100: 0.4/100 = 0.004.

Now, we need to find the decay constant, k. The decay constant, k, can be found using the formula:

k = (ln(Q₁/Q₀)) / t

Where:
- Q₁ is the final quantity, which is 0 since the drug starts to decay at the end of 0.35 seconds.
- t is the time duration, which is 0.35 seconds.

Substituting the values into the formula, we have:

k = (ln(0/0.3)) / 0.35

As we cannot take the natural logarithm of zero, this equation doesn't give us a valid value for k. It seems there might be an error in the problem statement or additional information is required.

Please check the problem statement or provide any additional data if available, so I can assist you further.