Answer: Just look at the explaintion
Explanation:
Part A:
The domain of the function C(t) represents the valid inputs, or possible values, for the variable t. In the context of the problem, the domain represents the time, t, in hours after injection. Since time cannot be negative in this context, the domain is all non-negative real numbers: t ≥ 0.
To find the range of the function C(t), we need to determine the possible values for the concentration, C. By analyzing the given function C(t) = −5t^2 + 10t, we can see that it is a quadratic function with a negative coefficient for the t^2 term. This means that the graph of the function opens downward, indicating that the concentration decreases over time.
Since the coefficient of the t^2 term is negative, the graph will have a maximum point. Therefore, the range of the function is the set of all real numbers less than or equal to the y-value of the maximum point.
Part B:
To determine the greatest concentration of the medication that a patient will have in their body and the time when that occurs, we can find the vertex of the quadratic function C(t) = −5t^2 + 10t. The vertex represents the maximum point of the graph.
The formula for finding the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b / (2a). In our case, a = -5 and b = 10.
x = -b / (2a) = -10 / (2(-5)) = -10 / (-10) = 1
Therefore, the time when the greatest concentration occurs is t = 1 hour.
To find the concentration at that time, we substitute t = 1 into the function C(t):
C(1) = −5(1)^2 + 10(1) = −5 + 10 = 5
Hence, the greatest concentration of the medication in the patient's body is 5 mg/L, and it occurs at t = 1 hour.