Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 30(a)

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MATH100 B December 2023

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Question 30(a)

A drug used to treat cancer is effective at low doses with an efficacy (ability to cure) that increases with the quantity of the drug used for treatment. However, at sufficiently high doses, the drug becomes toxic (can cause death). The fraction of patients surviving cancer with this drug treatment is given by $$S(x)={\frac {x}{k_{1}+x}}-{\frac {x}{k_{2}+x}}$$ where ${\textstyle x}$ is the drug quantity in hundreds of milligrams/day given to the patient and ${\textstyle 0<k_{1}<k_{2}}$. (a) Find ${\textstyle S(0)}$. Explain what your finding means in medical terms.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Substitute ${\textstyle x=0}$ directly into the survival function and see what happens to each term. Then ask yourself: In the context of the problem, what does a dosage of ${\textstyle x=0}$ mean medically, and how would that relate to patient survival?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

We can observe that plugging ${\textstyle x=0}$ into the formula for ${\textstyle S(x)}$ immediately gives $$S(0)={\frac {0}{0+k_{1}}}-{\frac {0}{0+k_{2}}}=0.$$ Mathematically, this tells us that the value of the survival function at ${\textstyle x=0}$ is ${\textstyle 0}$. In medical terms this means that without any drug treatment, no patients survive from the cancer (in the context of this particular drug treatment).

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