The concentration C of certain drug in a patient's bloodstream t hours after injection is given by

C(t)= t/3t^2+5

Required:
a. Find the horizontal asymptote of C(t). (Answer must be in slope-intercept form.)
b. Determins what happens to the concentration of the drug as t increases. As t increases, what value will c(t) approach.
c. Determine the time at which the concentration is highest.

Answer:

a) The horizontal asymptote of [tex]C(t)[/tex] is [tex]c = 0[/tex].

b) When t increases, both the numerator and denominator increases, but given that the grade of the polynomial of the denominator is greater than the grade of the polynomial of the numerator, then the concentration of the drug converges to zero when time diverges to the infinity. There is a monotonous decrease behavior.  

c) The time at which the concentration is highest is approximately 1.291 hours after injection.

Step-by-step explanation:

a) The horizontal asymptote of [tex]C(t)[/tex] is the horizontal line, to which the function converges when [tex]t[/tex] diverges to the infinity. That is:

[tex]c = \lim _{t\to +\infty} \frac{t}{3\cdot t^{2}+5}[/tex] (1)

[tex]c = \lim_{t\to +\infty}\left(\frac{t}{3\cdot t^{2}+5} \right)\cdot \left(\frac{t^{2}}{t^{2}} \right)[/tex]

[tex]c = \lim_{t\to +\infty}\frac{\frac{t}{t^{2}} }{\frac{3\cdot t^{2}+5}{t^{2}} }[/tex]

[tex]c = \lim_{t\to +\infty} \frac{\frac{1}{t} }{3+\frac{5}{t^{2}} }[/tex]

[tex]c = \frac{\lim_{t\to +\infty}\frac{1}{t} }{\lim_{t\to +\infty}3+\lim_{t\to +\infty}\frac{5}{t^{2}} }[/tex]

[tex]c = \frac{0}{3+0}[/tex]

[tex]c = 0[/tex]

The horizontal asymptote of [tex]C(t)[/tex] is [tex]c = 0[/tex].

b) When t increases, both the numerator and denominator increases, but given that the grade of the polynomial of the denominator is greater than the grade of the polynomial of the numerator, then the concentration of the drug converges to zero when time diverges to the infinity. There is a monotonous decrease behavior.  

c) From Calculus we understand that maximum concentration can be found by means of the First and Second Derivative Tests.

First Derivative Test

The first derivative of the function is:

[tex]C'(t) = \frac{(3\cdot t^{2}+5)-t\cdot (6\cdot t)}{(3\cdot t^{2}+5)^{2}}[/tex]

[tex]C'(t) = \frac{1}{3\cdot t^{2}+5}-\frac{6\cdot t^{2}}{(3\cdot t^{2}+5)^{2}}[/tex]

[tex]C'(t) = \frac{1}{3\cdot t^{2}+5}\cdot \left(1-\frac{6\cdot t^{2}}{3\cdot t^{2}+5} \right)[/tex]

Now we equalize the expression to zero:

[tex]\frac{1}{3\cdot t^{2}+5}\cdot \left(1-\frac{6\cdot t^{2}}{3\cdot t^{2}+5} \right) = 0[/tex]

[tex]1-\frac{6\cdot t^{2}}{3\cdot t^{2}+5} = 0[/tex]

[tex]\frac{3\cdot t^{2}+5-6\cdot t^{2}}{3\cdot t^{2}+5} = 0[/tex]

[tex]5-3\cdot t^{2} = 0[/tex]

[tex]t = \sqrt{\frac{5}{3} }\,h[/tex]

[tex]t \approx 1.291\,h[/tex]

The critical point occurs approximately at 1.291 hours after injection.

Second Derivative Test

The second derivative of the function is:

[tex]C''(t) = -\frac{6\cdot t}{(3\cdot t^{2}+5)^{2}}-\frac{(12\cdot t)\cdot (3\cdot t^{2}+5)^{2}-2\cdot (3\cdot t^{2}+5)\cdot (6\cdot t)\cdot (6\cdot t^{2})}{(3\cdot t^{2}+5)^{4}}[/tex]

[tex]C''(t) = -\frac{6\cdot t}{(3\cdot t^{2}+5)^{2}}- \frac{12\cdot t}{(3\cdot t^{2}+5)^{2}}+\frac{72\cdot t^{3}}{(3\cdot t^{2}+5)^{3}}[/tex]

[tex]C''(t) = -\frac{18\cdot t}{(3\cdot t^{2}+5)^{2}}+\frac{72\cdot t^{3}}{(3\cdot t^{2}+5)^{3}}[/tex]

If we know that [tex]t \approx 1.291\,h[/tex], then the value of the second derivative is:

[tex]C''(1.291\,h) = -0.077[/tex]

Which means that the critical point is an absolute maximum.

The time at which the concentration is highest is approximately 1.291 hours after injection.

helppp meeeeeeeeeeeeeeeeee kdndnd

Mr. Cornapple is the fruit & vegetable buyer for a large school district. In a given week, he has a budget of $6,550. Bushels of apples cost $40 each, and bushels of corn cost $30 each. The following inequality represents the amount of money that Mr. Cornapple spends on apples and corn.

I am trying to find an explanation for this algebra 7th grade math problem 3+x>-9

Order of operation with Integers

Find the equation of the circle with the given characteristics.

Center: (2, −2) Radius: 2
The solution is:

What does E stand for?

Which of terms could be added to √5?

what is 7/9 ÷ 1/3 plz help ​

please help me)

question:
choose the meaning that gives a negative connotation of the word "snake"

A: scaly
B: limbless
C: poisonous
D: calm

(all I know is it's not D) please help me with answering this, it's pretty easy, but I am kinda stuck on what would make the most sense. ​

29/3 as an a mixed number

Identify which theorem will prove that

There were 28 children in the final of the Air Guitar competition. If 25% of the finalists were girls, who
many boys were there in the final?

the letters n and d represent integers greater than 0. Which of the following is the same as -(n/d)

how many 3/10 size are in 3 wholes​

In a local city the temperature at noon is -3 Celsius the temperature is expected to decrease by 2.5 sell each hour until the temperature reaches -18 Celsius how many hours will it take for the temperature to reach -18 Celsius

brazill is beststst idiotttefwweffsfaffsgjhfgsadhgvfgdsfghds fsdfsdfsd trava zap trava zap

What is the value of P(C or T)?

A.

B.

C.

D

PLEASE HELP!!!!! 1-Suppose the population of a city is 50000 and growing 3% each year.
What is the population after 25 years?
20 points
A)104689
B)107250
C)99500000
D)12.

6*9+(6+9)=?

Plz I really need this in the next minute.

quentin is filling a glass that holds 1 3/4 cups of water. he is using 1/4 cup measuring cup. how many times will he have to fill the smaller measuring cup to equal 1 3/4 cups?

Is 7/15 reapeating or terminating

IS IT 18, OR 27 ON EDGE

What is the equation of the line that passes through the point (5, - 7) and has a slope of

triangle has one angle 33 degrees, 114 degrees and x degrees. What is the value or x

PLEASE HELP ASAP NOW!!!!!!!!!!!!!!

HELP! please! ASAP! pleasee

Which of the following statements are true? Select all that apply.

1.5 tenths x 20 = ______ tenths

5/3 ÷ 7 PLZ HELP ME OUT!!! (I can't figure it out!)

nancy earns $12.50 an hour cleaning houses, but doesn’t work on saturday or sunday. if she works from 8:00 am to 2:00 pm, how much money will she make ?