A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time t >= 0(in hours since giving the drug) the concentration is given by(in mg/L)
c(t) = 5t / (t^2+1)
Graph the function.
From the graph, how do I find
a)the highest concentration of drug that is reached in the patient's bloodstream
b)what happens to the drug concentration after a long period of time.
c)How long does it take for the concentration to drop below 0.3 mg/L
TIA
The highest will be the highest on the graph, the peak.
What happens as t approaches infinity?
c) read the graph.
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To graph this function, you can use software or online tools like Desmos or any graphing calculator. Here is a step-by-step explanation of how to find the answers to the questions:
a) To find the highest concentration of the drug reached in the patient's bloodstream, you need to locate the peak point on the graph. This is the point where the concentration is the highest.
1. Plot the graph of the function c(t) = 5t / (t^2+1).
2. Identify the highest point on the graph, which corresponds to the peak concentration.
b) To determine what happens to the drug concentration after a long period of time, you need to analyze the behavior of the graph as t approaches infinity.
1. Examine how the graph behaves as t becomes larger and larger.
2. If the concentration approaches a specific value (e.g., 0), then it indicates that the drug concentration tends towards that value after a long period of time.
3. Analyze the trend of the graph to infer what happens to the drug concentration after a long period of time.
c) To determine how long it takes for the concentration to drop below 0.3 mg/L, you need to locate the point(s) where the concentration equals 0.3 mg/L on the graph.
1. Plot the graph of the function c(t) = 5t / (t^2+1).
2. Identify the point(s) on the graph where the concentration is 0.3 mg/L.
3. Determine the corresponding time value(s) when the concentration reaches 0.3 mg/L.
Remember, graphing the function will allow you to visually analyze its behavior, which will help you answer these questions more accurately.
To graph the function c(t) = 5t / (t^2+1), you can use a graphing calculator or software, or manually plot points on a coordinate plane. Here are the steps to manually plot points and graph the function:
1. Choose several values for t, such as -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.
2. Calculate the corresponding values of c(t) using the function. For example, if t = 1, c(1) = 5(1) / (1^2+1) = 5/2.
3. Plot the points (t, c(t)) on a coordinate plane.
4. Connect the points using a smooth curve to create the graph of the function.
To find the highest concentration of the drug that is reached in the patient's bloodstream, look for the peak point on the graph. This is the point where the concentration is highest.
To determine what happens to the drug concentration after a long period of time, observe the behavior of the graph as t approaches infinity. If the graph approaches a specific value or becomes asymptotic to a particular line, that indicates the long-term behavior of the drug concentration.
To find out how long it takes for the concentration to drop below 0.3 mg/L, look for the point on the graph where the concentration (c(t)) is equal to or less than 0.3 mg/L. Find the corresponding value of t at that point.