SOLUTION: The concentration of a drug in a bloodstream, measured in milligrams per liter, can be modeled by the function, C(t)=(12t+4)/(3tē+2), where t is the number of minutes after injecti

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Question 977588: The concentration of a drug in a bloodstream, measured in milligrams per liter, can be modeled by the function, C(t)=(12t+4)/(3tē+2), where t is the number of minutes after injection of the drug. When will the drug be at its highest concentration? Approximate your answer rounded to two decimal places.
Found 2 solutions by Fombitz, Boreal:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the derivative,
dC%2Fdt=%28%283t%5E2%2B2%2912-%2812t%2B4%29%286t%29%29%2F%283t%5E2%2B2%29%5E2
dC%2Fdt=%2836t%5E2%2B24-72t%5E2-24t%29%2F%283t%5E2%2B2%29%5E2
dC%2Fdt=%28-36t%5E2-24t%2B24%29%2F%283t%5E2%2B2%29%5E2
dC%2Fdt=%28-12%283t%5E2%2B2t-2%29%29%2F%283t%5E2%2B2%29%5E2
Set the derivative equal to zero.
3t%5E2%2B2t-2=0
t%5E2%2B%282%2F3%29t-2%2F3=0
t%5E2%2B%282%2F3%29t%2B1%2F9=2%2F3%2B1%2F9
%28t%2B1%2F3%29%5E2=7%2F9
t%2B1%2F3=0+%2B-+sqrt%287%29%2F3
t=-1%2F3+%2B-+sqrt%287%29%2F3
Only use the positive time value.
t=-1%2F3+%2B+sqrt%287%29%2F3

Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
(12t+4)/(3tē+2)
Graphing gives an idea of the maximum, and I would use a graphing calculator on this and get t=0.55 giving a maximum of 3.65.
Of note is that the first derivative set to zero gives 2/3 with a maximum of 3.60.


graph%28300%2C300%2C-3%2C3%2C-10%2C10%2C%2812x%2B4%29%2F%283x%5E2%2B2%29%29