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 C(t) = −2t^2 + 8t, where the time, t, is hours after injection.

Part A: What are the domain and range of the function C(t) based on the context of the problem? Show all necessary calculations. (5 points)

Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body. (5 points)

Answer 1

Part A:

The domain of the function C(t) is the set of all possible values of t for which the function is defined and makes sense in the context of the problem. In this case, since the time t represents hours after injection, the domain of C(t) must be non-negative (i.e., t ≥ 0) since negative values of t do not make sense in the context of the problem. Therefore, the domain of C(t) is:

Domain: t ≥ 0

The range of the function C(t) is the set of all possible values of C that the function can take. Since the function is a quadratic function of t with a negative coefficient on the t^2 term, the function will have a maximum value at the vertex of the parabola. To find the range of the function, we need to find the maximum value of the function.

Part B:

To find the greatest concentration of the medication that a patient will have in their body, we need to find the maximum value of the function C(t). The maximum value of the function occurs at the vertex of the parabola, which is given by the formula:

t = -b/2a

where a is the coefficient of the t^2 term, b is the coefficient of the t term, and c is the constant term. In this case, a = -2, b = 8, and c = 0, so we have:

t = -b/2a = -8/(2*(-2)) = 2

Therefore, the maximum concentration of the medication occurs 2 hours after injection. To find the maximum concentration, we substitute t = 2 into the function:

C(2) = -2(2)^2 + 8(2) = 8

So the greatest concentration of the medication that a patient will have in their body is 8 mg/L.

To graph the function, we can plot some points and draw a smooth curve through them:

*Table Attached*

The vertex of the parabola is at (2, 8), which is the maximum point on the graph. The graph of the function is a downward-opening parabola that passes through the points (0, 0) and (4, 0).