# Answer in Calculus for Artika #261828

A mathematical biologist created a model to administer medicine reaction (measured in change of blood pressure or temperature) with the model given by R=m^2(c/2-m/3) where c is a positive constant and m is the amount of medicine absorbed into the blood. The sensitivity to the medication is defined to be the rate of change of reaction R with respect to the amount of medicine m absorbed in the blood.

i. Find the sensitivity, R’.

(ii) Find the amount of medicine that is being absorbed into the blood when the reaction is maximum.

(iii) Find the instantaneous rate of change of sensitivity with respect to the amount of medicine absorbed in the blood.

(iv) Find the amount of medicine that is being absorbed into the blood when the sensitivity is maximum.

Given “R=m^{2}=left(frac{c}{2}-frac{m}{3}right)”

(i) Sensitivity of medicine “R=m^{2}left(frac{c}{2}-frac{m}{3}right)”

Let “u=m^{2}, quad v=frac{c}{2}-frac{m}{3}.”

Sensitivity “=frac{d R}{d M}=frac{d}{d m}left(m^{2}left(frac{c}{2}-frac{m}{3}right)right)”

We know that “(uv) =u^{prime} v+v u^{prime}”

“begin{aligned} frac{d R}{d m} &left.=frac{d}{d m}left(m^{2}right)left(frac{c}{2}-frac{m}{3}right)+frac{p^{2}}{2} d m^{2} cdot frac{d}{d m}left(frac{c}{2}-frac{m}{3}right)right) \ &=2 mleft(frac{c}{2}-frac{m}{3}right)+m^{2}(-1 / 3) \ &=m c-frac{2 m^{2}}{3}-frac{m^{2}}{3} \ &=mc-m^{2} \ text { Sensitivity } &=m c-m^{2} end{aligned}”

(ii) When reaction is maximum “frac{d R}{d m}=0”

“mc-m^{2}=0nn\ mc=m^{2} nn\therefore c=m”

c is the amount of medicine of absorbed into blood.

(iii) Instantaneous rate of sensitivity

“begin{aligned}nnfrac{d s}{d m} &=frac{d}{d m}left(m c-m^{2}right) \nn&=frac{d}{d m}(m c)-frac{d}{d m}left(m^{2}right) \nn& =c-2 mnnend{aligned}”

Instantaneous rate of sensitivity “=c-2 m”

(iv)

“begin{aligned}nn&text { Sensitivity = maximum, } frac{d s}{d m}=0 \nn&begin{array}{c}nnfrac{d s}{d m}=0 \nnc-2 m=0 \nnc=2 m \nnm=frac{c}{2}nnend{array}nnend{aligned}”

Hence, when “m=frac{c}{2},” is absorbed when sensitivity is maximum.