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$P, Q, R$ and $S$ are four types of dangerous microbes recently found in a human habitat. The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within $24$ hours of entering the body. The danger to human beings varies proportionately with the toxicity, potency and growth attributed to a microbe shown in the figure below:

A pharmaceutical company is contemplating the development of a vaccine against the most dangerous microbe. Which microbe should the company target in its first attempt?

1. $P$
2. $Q$
3. $R$
4. $S$
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As per the question, it is quite clear that the danger of a microbe to human being will be directly proportional to potency and growth. At the same time it is inversely proportional to toxicity, defined as( more dangerous will a microbe be if lesser of its milligram is required).

So, $\text{Level Of Danger (D)} \propto \text{Growth (G)} \propto \text{Potency (P) }\dfrac{1}{\propto} \text{ Toxicity (T)}$

$D = \dfrac{KGP}{T}$

where K is contant of proportionality.

So level of danger will be maximum for $S.$

Given by,

$DS =\dfrac{0.8\times \large\pi(10)^{2}}{200}$

$\qquad= 1.256$

Similar calculations for $D{_P} , D{_Q} , D{_R}$ can be done. Which will consequently lead to $D{_S}$ being the most dangerous and hence will be targeted first.

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Can some one explain the following statement from question?

The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within 24 hours of entering the body

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it just represents the growth rate - how fast a microbe grows.
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I don't know whether My way of working is correct

but what my intuitive analysis is

danger is based on 3 factors

(1)toxicity

(2)potency

(3)growth

since if we see given graph clearly S is most toxic as less quantity of S is required to more damage(indicated by what is written on graph's y axis)

S has more chance than P, Q and R surviving the human immunity system

but growh rate of S is comparatively slower than that of P, Q and R.

but since S is ahead of P, Q and R by 2 factors so overall danger will be possessed by S most.

please let me know if its correct.
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It's clearly mentioned in the question that the danger to human beings varies proportionately with the toxicity, potency and growth. Then why you're taking the inverse of the toxicity?
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but we have to think generally its given 800 mg of P is required for burning down some wieght but S needs only 200mg to do the same work so S is more dangerous
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Why we are only multiplying all the factors ....why shouldn' t we add them?
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this apparoch worked due to fact that here data can be easily distinguished. But if data have been more compact or less degree of change then then accurate calculation would have been required.
danger is inversely proportional to toxicity as the most dangerous would be the one which require less milligram to destroy half of body mass

SO, danger = (K * potency * growth)/toxicity

SO OPTION (D)
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$\text{danger to human beings} \propto \text{toxicity}\\ \indent \qquad\qquad\qquad\qquad\qquad\,\,\,\propto \text{potency}\\ \indent \qquad\qquad\qquad\qquad\qquad\,\,\,\propto \text{growth}$

$\implies \text{danger}(x) = c \times \text{toxicity}(x) \times \text{potency}(x) \times \text{growth}(x)$

So,

$\text{danger}(P) = c \times \underbrace{\text{toxicity}(P)}_{800} \times \underbrace{\text{potency}(P)}_{0.4} \times \underbrace{\text{growth}(P)}_{50}\\\\ \indent\qquad\qquad \,\,\,= 16000 c$

$\text{danger}(Q) = c \times 600 \times 0.5 \times 40= 12000 c$

$\text{danger}(R) = c \times 300 \times 0.4 \times 30= 3600 c$

$\text{danger}(S) = c \times 200 \times 0.8 \times 20= 3200 c$

Since $P$ poses the most danger, the company should target $P$ first.

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It is wrong. Check the answer below. You did a bit mistake while considering the toxicity.
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Yes I see my mistake now.

Thanks for pointing that out :)

Tnx for the answer....As the question says- The danger to human beings varies proportionately with the toxicity, potency and growth...nothing about direct proportionality

Hence, Toxicity is considered as inversely proportional as given in the graph.

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So where it is mentioned that toxicity has to be inversely proportional
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They said varies proportionally but not is proportional to. To be frank I don't understand why the paper setters care so much about skills in interpreting the question rather than the analytical and reasoning skills of the student. I would have surely considered it as directly proportional the moment I see varies proportionally.

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