Will someone please help me with the math question below and show the calculations so I can get a better grip on the concept.

Question

Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function  where the time, t, is hours after injection.

Part A: What is the domain of the function C(t) based on the context of the problem? Show all necessary calculations.

Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body.

Answer 1

Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function  where the time, t, is hours after injection.

Part A: What is the domain of the function C(t) based on the context of the problem? Show all necessary calculations.

Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body.

Hi Taylor,

The keywords in the problem are as follows:

based on the context of the problem tells us that we should treat the variables t and C(t) not as abstract concepts, but as some measurable values with physical meaning(s)

concentration tells us that C(t) ≥ 0

after tells us that t > 0

Now we need to solve the inequality ≥ 0

For t > 0  the denominator t² + 4t +3 ≥ 3 > 0

Therefore in order for C(t) to be positive (or zero) the numerator has to be positive (or zero) or

-10 t² + 50t  ≥ 0

Dividing both sides by 10 t (we can do it, because t > 0) we obtain

- t + 5  ≥ 0    

The next step is to add t to both sides of the inequality

5 ≥ t

Hence the domain of C(t) is 0 < t ≤ 5

From the graph you can see that C(t) has a maximum at C (1 hour) = 5 mg/L

Answer 2

Hi Taylor,

Part A

(a): The domain of the function

Because the medication concentration C(t) is a function of time (t), the domain is the possible values t can take.

t, cannot take negative values (i.e. it is not possible to have a negative time).

The least possible value of t is 0

So, the domain of the function based on the context is:

Part B:

You need to factorize and find the intervals we get (1, 5], however, the best method can be plotting the graph.

From the graph, the maximum y-value is 5.

Hence, the greatest concentration of the medication is 5 mg/L