A medicine in the bloodstream has a concentration of c(t) =at/t^2 + b where a=3 and b=1
Approximate the highest concentration of the medicine reached in the bloodstream
Find algebraically when c(t) less then Sign 0.5
The question does not make sense.
a t / t ^ 2 =
3 t / t ^ 2 =
3 t / ( t * t ) =
3 / t
a t / t ^ 2 + b =
3 t / t ^ 2 + 1 =
3 / t + 1
Tends to infinity as x tends towards 0.
How would i approximate the highest concentration reached in the bloodstream
Not at all.
highest concentration = infinity as x tends towards 0.
The question also states to determine how long it takes for the medicine to drop below 0.2 how could i do this
Rose, it was pointed out to you by Bosnian that your question makes no sense, yet you keep asking questions pertaining to it.
your equation is
c(t) = 3t/t^2 + 1
which reduces to
c(t) = 3/t + 1
Even at the beginning, when t=0, this would be undefined.
That is, the concentration would be infinitely huge, rather silly, don't you think?
the graph looks like this:
http://www.wolframalpha.com/input/?i=plot+y+%3D+3%2Fx+%2B+1
As t gets larger, c(t) will approach 1
for c(t) < .5
3/t < .5
3 < .5t
t > 6
the graph confirms this
To find the highest concentration of the medicine reached in the bloodstream, we need to find the maximum value of the function c(t) = at/t^2 + b.
To do this, we can use calculus. We need to find the critical points of the function, which are the values of t where the derivative of c(t) is equal to zero. Let's first find the derivative of c(t):
c'(t) = [(a*t^2 + b)*(2*t) - (a*t)*(2*t)] / (t^2)^2
= (2*a*t^3 + 2*b*t - 2*a*t^3) / t^4
= (2*b*t) / t^4
= 2b/t^3
Now, let's set c'(t) equal to zero and solve for t:
2b / t^3 = 0
Since b is given as 1, we have:
2 / t^3 = 0
There is no solution to this equation since a fraction divided by zero is undefined. So, there are no critical points for c(t).
To determine whether the function has a maximum or minimum, we need to analyze the behavior of c(t) as t approaches infinity and negative infinity.
As t approaches infinity, both the numerator and denominator of c(t) become very large positive numbers. So, we can conclude that the limit of c(t) as t approaches infinity is 0 (zero).
As t approaches negative infinity, both the numerator and denominator of c(t) become very large negative numbers. However, since t^2 is always positive, the limit of c(t) as t approaches negative infinity is also 0 (zero).
Since c(t) approaches zero as t approaches both positive and negative infinity, we can conclude that c(t) has no maximum or minimum value. Therefore, there is no highest concentration of the medicine reached in the bloodstream.
Moving on to the second part of your question: To find when c(t) is less than 0.5 algebraically, we need to solve the inequality c(t) < 0.5. Let's rewrite the equation:
at/t^2 + b < 0.5
Multiply both sides of the inequality by t^2:
at + b*t^2 < 0.5*t^2
Rearrange the terms:
b*t^2 - 0.5*t^2 + at < 0
Combine like terms:
(0.5*a - b)*t^2 + at < 0
To solve this inequality algebraically, we need more information about the constants a and b. Please provide the values of a and b so that we can continue.