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?

Danger is directly proportional to toxicity, potency, and growth:

From the y-axis description, the measure given is *"milligrams of microbe required to destroy half of the body mass in kilograms"* ($LD_{50}$). A lower required mass means a higher toxicity, so:

The growth is proportional to the area of the circle, which is proportional to $(\text{diameter})^2$:

Thus, the danger metric for each microbe is proportional to $\frac{\text{Potency} \times d^2}{\text{Mass required}}$:

• Microbe P:

  $$\text{Danger}(P) \propto \frac{0.4 \times 50^2}{800} = \frac{0.4 \times 2500}{800} = 1.25$$

• Microbe Q:

  $$\text{Danger}(Q) \propto \frac{0.5 \times 40^2}{600} = \frac{0.5 \times 1600}{600} \approx 1.33$$

• Microbe R:

  $$\text{Danger}(R) \propto \frac{0.4 \times 30^2}{300} = \frac{0.4 \times 900}{300} = 1.20$$

• Microbe S:

  $$\text{Danger}(S) \propto \frac{0.8 \times 20^2}{200} = \frac{0.8 \times 400}{200} = 1.60$$

Since microbe S has the highest danger index ($1.60$), the pharmaceutical company should target S first.