The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is C=3t^2+t/ t^3+50
A. Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem.
B. approximate the time when the bloodstream concentration is greatest.
A.) Horizontal Asymptote- y=0, you can never have more than 1 (100%) or less than 0 (0%) of the chemical in the bloodstream
B.) 46% is the maximum after 4.5 hours
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.
These are the answers your looking for, hope this helps ;)
To find the horizontal asymptote of the function C(t), we need to analyze the behavior of the function as t approaches positive or negative infinity.
A. To determine the horizontal asymptote:
1. Take the highest power of t in the numerator and the denominator, which in this case is t^3.
2. Divide each term in the function by t^3.
C(t) = (3t^2 + t) / (t^3 + 50)
C(t) / t^3 = (3t^2 / t^3) + (t / t^3) / (t^3 / t^3 + 50 / t^3)
Simplifying further, we get:
C(t) / t^3 = 3/t + 1/t^2 / (1 + 50 / t^3)
As t approaches infinity, both 3/t and 1/t^2 tend to zero. Therefore, the entire expression (C(t) / t^3) approaches zero.
This indicates that the horizontal asymptote of the function C(t) is y = 0. In the context of the problem, it means that as time goes on (approaching infinity), the concentration of the chemical in the bloodstream approaches zero. This suggests that the chemical is gradually eliminated from the bloodstream.
B. To approximate the time when the bloodstream concentration is greatest:
To find the maximum value of a function, we need to first find its critical points.
1. Take the derivative of the function C(t) with respect to t:
C'(t) = [6t + 1(t^3 + 50) - (3t^2 + t)(3t^2)] / (t^3 + 50)^2
Simplifying further:
C'(t) = (6t + t^4 + 50 - 9t^4) / (t^3 + 50)^2
C'(t) = (t^4 - 3t^2 + 50) / (t^3 + 50)^2
2. Set C'(t) equal to zero and solve for t to find the critical points:
t^4 - 3t^2 + 50 = 0
This is a quadratic equation in t^2. Solve it and find the values of t.
Once you have the critical points, you can evaluate the function C(t) at those points to determine the maximum concentration. The time value associated with the maximum concentration is the approximate time when the bloodstream concentration is greatest.