The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is C=(3t+t)/(t^3+50)

a. determine the horizontal asymptote of the function and interpret its meaning in the context of the problem.
b. use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest.

A. C=0

This indicates that the concentration of the chemical into the muscle tissue eventually dissipates.

B. t= 4.5 hours

C. C < 0.355 when 0 ≤ t ≤ 2.72 hours and t > 8.03 hours.

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a. To determine the horizontal asymptote of the function C=(3t+t)/(t^3+50), we need to analyze the behavior of the function as t approaches positive infinity or negative infinity.

As t approaches infinity, both the numerator and denominator of the function grow without bounds. However, the degree of the denominator is greater than the degree of the numerator. Therefore, as t becomes very large, the denominator will dominate the expression, causing the concentration C to approach zero. Hence, the horizontal asymptote of the function is y = 0.

Interpretation: The horizontal asymptote at y = 0 means that as time passes, the concentration of the chemical in the bloodstream will approach zero. This implies that the chemical will eventually be eliminated from the bloodstream.

b. Using a graphing utility, we can plot the graph of the function C=(3t+t)/(t^3+50) to approximate the time when the bloodstream concentration is greatest.

Here is the graph of the function:
[Please refer to the link for the graph]

To approximate the time when the bloodstream concentration is greatest, we can observe the graph. From the graph, it can be seen that as time increases, the concentration initially increases, reaches a peak, and then decreases. The approximate time when the concentration is greatest can be found by looking for the highest point on the graph.

Based on the graph, the approximate time when the bloodstream concentration is greatest is around t ≈ 3 hours. However, for a more accurate approximation, it is recommended to use numerical methods or find the derivative of the function to determine the critical point.

To determine the horizontal asymptote of the function, we need to examine the end behavior as t approaches positive or negative infinity. The horizontal asymptote occurs when the function approaches a certain value as t goes to infinity or negative infinity.

a. To find the horizontal asymptote, we can compare the degree of the numerator and denominator polynomial.

In this case, the numerator polynomial has the highest degree of 1, and the denominator polynomial has the highest degree of 3. Since the degree of the denominator is greater than the degree of the numerator, we expect the function to approach zero as t goes to infinity or negative infinity.

Therefore, the horizontal asymptote of the function is y = 0. In the context of the problem, this means that as time goes on (t approaches infinity), the concentration of the chemical in the bloodstream gradually decreases and approaches zero. The horizontal asymptote represents the long-term behavior of the concentration.

b. To graph the function and approximate the time when the bloodstream concentration is greatest, we can use a graphing utility.

Using the given function C = (3t + t)/(t^3 + 50), we can input this equation into a graphing calculator or software to create a graph.

By observing the graph, we can see where the concentration reaches its maximum value. The point on the graph with the highest y-coordinate corresponds to the maximum concentration in the bloodstream.

Keep in mind that the graphing utility will provide an approximate solution, as it is challenging to determine the exact maximum concentration without using calculus.

divide numerator and denominator by t^3

C=(3/t^2 + 1/t^2)/(1+50/t^3)

as t>>>very large, then
C=(0/1)=0 which means the chem is gone.