# The concentration of a drug in a patient's bloodstream t hours after it is taken after it is...

## Question:

The concentration of a drug in a patient's bloodstream t hours after it is taken after it is taken is given by {eq}C(t) = \frac{0.016t}{(t+2)^2} mg/cm^3 {/eq}

Find the maximum concentration of the drug and the time at which it occurs.

## Calculus

A mathematical study that helps to represent the system function or equation in infinitesimals rate changes or absolute value is known as calculus. It has two branches; one is differential, and the other is integration.

Given Data

• The concentration of drug is: {eq}C\left( t \right) = \dfrac{{0.016t}}{{{{\left( {t + 2} \right)}^2}}}\;{\rm{mg/c}}{{\rm{m}}^3} \cdots\cdots\rm{(I)} {/eq}

Differetiate the concentration of drug function with respect to time is

{eq}\begin{align*} \dfrac{{d\left( {C\left( t \right)} \right)}}{{dt}} &= C'\left( t \right)\\ C'\left( t \right) &= \dfrac{{d\left( {\dfrac{{0.016t}}{{{{\left( {t + 2} \right)}^2}}}} \right)}}{{dt}}\\ &= \dfrac{{0.016{{\left( {t + 2} \right)}^2} - 2\left( {t + 2} \right)0.016t}}{{{{\left( {{{\left( {t + 2} \right)}^2}} \right)}^2}}}\\ &= \dfrac{{0.016\left( {t + 2} \right) - 0.032t}}{{{{\left( {t + 2} \right)}^3}}} \end{align*} {/eq}

To find the maximum value the derivative must equal to zero.

For maximum concentration of the drug the derivative of concentration of drug equate to zero.

{eq}\begin{align*} C'\left( t \right) = \dfrac{{d\left( {\dfrac{{0.016t}}{{{{\left( {t + 2} \right)}^2}}}} \right)}}{{dt}} &= 0\\ \dfrac{{0.016\left( {t + 2} \right) - 0.032t}}{{{{\left( {t + 2} \right)}^3}}} &= 0\\ 0.016\left( {t + 2} \right) - 0.032t &= 0\\ 0.016t + 0.032 - 0.032t &= 0\\ 0.016t &= 0.032\\ t &= 2 \end{align*} {/eq}

Thus the time at which maximum concentration of drug occur is {eq}2\;{\rm{s}} {/eq}

Subsitute and solve the expression (I)

{eq}\begin{align*} C\left( t \right) &= \dfrac{{0.016\left( 2 \right)}}{{{{\left( {2 + 2} \right)}^2}}}\;{\rm{mg/c}}{{\rm{m}}^3}\\ &= 0.002\;{\rm{mg/c}}{{\rm{m}}^3} \end{align*} {/eq}

Thus the maximum concentration in patient's bloodstream is {eq}0.002\;{\rm{mg/c}}{{\rm{m}}^3} {/eq}