To solve the problem, we need to determine the dosage(s) [tex]\( x \)[/tex] that would make the sensitivity [tex]\( S \)[/tex] equal to zero.
### Part a: Finding the Dosages that Give Zero Sensitivity
1. Set the sensitivity function [tex]\( S \)[/tex] to zero:
Given the sensitivity function [tex]\( S = 380x - x^2 \)[/tex], we set [tex]\( S \)[/tex] to zero:
[tex]\[
380x - x^2 = 0
\][/tex]
2. Factor the equation:
To solve the equation [tex]\( 380x - x^2 = 0 \)[/tex], we can factor out [tex]\( x \)[/tex]:
[tex]\[
x(380 - x) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
The factored equation gives us two possible solutions for [tex]\( x \)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad 380 - x = 0
\][/tex]
4. Solve the second equation:
[tex]\[
380 - x = 0 \implies x = 380
\][/tex]
Thus, the dosages that give zero sensitivity are [tex]\( x = 0 \)[/tex] ml and [tex]\( x = 380 \)[/tex] ml.
So, the correct choice for part (a) is:
A. The dosage(s) would give zero sensitivity when [tex]\( x = 0 \)[/tex] ml, 380 ml.
### Part b: Explanation of the Result
The dosages that give zero sensitivity are [tex]\( x = 0 \)[/tex] ml and [tex]\( x = 380 \)[/tex] ml. These results have meaningful interpretations:
- [tex]\( x = 0 \)[/tex] ml: This means that if no drug is administered (the dosage is zero), the sensitivity to the drug is also zero. This is logical because, without any drug, there is nothing to evoke a sensitivity response.
- [tex]\( x = 380 \)[/tex] ml: This indicates that there is a specific dosage (380 ml) at which the sensitivity to the drug again drops to zero. This suggests that at this high dosage, the drug's effectiveness ceases, potentially due to over-saturation or the body's tolerance limit being reached.
In summary, the dosages that result in zero sensitivity are [tex]\( x = 0 \)[/tex] ml (no drug administered) and [tex]\( x = 380 \)[/tex] ml (a high dosage where the sensitivity effect is neutralized or nullified).
