Science:Math Exam Resources/Courses/MATH101 C/April 2025/Question 01 (c)

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MATH101 C April 2025

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q9  •  Q10  •  

   Other MATH101 C Exams

  •  April 2024  •  

Question 01 (c)

A certain population of wild salmon swims upstream, past a research station. A biologist keeps track of how many individual salmon pass the station over time. Suppose $\underset{a}{\overset{b}{\int }}q(t)dt$ gives the number of salmon that passed from time $t=a$ to time $t=b$, where $t$ is measured in hours. Select the best interpretation of $q(t)$ from the list below. A: $q(t)$ gives the total distance (measured in kilometres) the salmon have travelled from time $t=a$ to time $t=b$. B: $q(t)$ gives the acceleration of the salmon, from time $t=a$ to time $t=b$. C: $q(t)$ gives the number of salmon that passed the station from time $t=a$ to time $t=b$, where $t$ is measured in hours. D: $q(t)$ gives the rate (in number of salmon per hour) at which salmon pass the research station at time $t$. E: $q(t)$ gives the rate (in kilometres per hour) at which salmon pass the research station at time $t$. F: $q(t)$ gives the rate of change (in salmon per hour per hour, $\frac{\text{salmon}}{{\text{hour}}^2}$) of the rate at which salmon are passing the research station.

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
What does the fundamental theorem of calculus say? Note that the upper bound of the integral defining $F$ is $x^2$, not $x$.

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Solution
Writing the total number of salmon that have passed through the station up until time $t$ as $Q(t)=\underset{a}{\overset{t}{\int }}q(s)ds$, we see that the rate at which salmon pass through the station at time $t$ should be given $\frac{dQ}{dt}$. But by the fundamental theorem of calculus, $\frac{dQ}{dt}=q(t)$, so the answer must be D.

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