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A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?

Can this question be used to assess a student's understanding of calculus?

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    $\begingroup$ Calculus allows you to find the rate at which that distance is increasing. Logic (no calculus needed) tells you that it's increasing. The scenario is fine. The question (increasing or decreasing) makes no sense to me. $\endgroup$
    – Sue VanHattum
    Oct 17, 2023 at 19:08
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    $\begingroup$ "Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?" -- This sounds like a reasonable "get your bearings" first step, but no calculus is involved yet. Part 2 might ask for the distance between the person and the balloon as a function of time $t$. Then part 3...etc. As it sits (part 1 only), I don't think it can be used to assess understanding of calculus. $\endgroup$
    – Nick C
    Oct 17, 2023 at 19:13
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    $\begingroup$ @Sue VanHattum: The question (increasing or decreasing) makes no sense to me. --- I just saw this question and was wondering about the question myself. At first I thought maybe the question was not worded as intended, and maybe the question was whether the speed the balloon is moving away from the person is increasing or decreasing, but a quick sketch (right triangle with legs $5t$ and $12t)$ shows that this speed is a constant $13$ feet per second, which incidentally I didn't need calculus to determine. This all assuming that the person is walking at a constant speed of $5$ ft/sec . . . $\endgroup$ Oct 17, 2023 at 19:15
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    $\begingroup$ No offence intended, but I think this example is a good illustration of several reasons why so many students dislike math: To illustrate to them why math is "important" and how it can be "applied" we give them a "real-world problem" which (a) is completely contrived, (b) has a solution which is plain obvious, no matter whether you know any math at all (let alone calculus), and (c) has a pointless "gotcha" in it ("Why would it matter whether you consider the situation at 4 seconds or at 1 or at 100?" - "Yepp, it doesn't - ha!") $\endgroup$ Oct 17, 2023 at 22:39
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    $\begingroup$ @JochenGlueck Well, deriving the relative motion from two known trajectories is both a "real world" and, at least occasionally, a "calculus" problem though, of course, straight line constant speed motion needs nothing beyond elementary 2 or 3D vector geometry and the setup the OP proposed is ridiculously simple even for that. So, if one wants this scenario, I would set up two variable speed curvilinear motions instead (nothing too fancy: 2 stones thrown into the air at an angle by Alice and Bob would do) and ask the students to use physics, geometry, and calculus to answer a few queries. :-) $\endgroup$
    – fedja
    Oct 18, 2023 at 1:57

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For me this is a question not for calculus but for right triangles, Pythagoras and not really a calculus question.

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Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?

The balloon and the person are always moving further away from each other (balloon moving further away vertically, person moving further away horizontally), so it's just common sense that the distance between them is always increasing.

That said, you might see this sort of question being asked as a "first step" in a calculus question consisting of multiple parts (for the purpose of scaffolding the student through the problem-solving procedure):

(i) Is the distance between the person and the balloon increasing or decreasing after they start moving?

(ii) Draw triangle with the following sides labeled: the height of the balloon, the distance of the person from the balloon launch site, and the distance between the person and the balloon.

(iii) Set up an equation relating the sides of that triangle.

(iv) Use implicit differentiation to differentiate the equation with respect to time.

(v) How fast is the distance between the person and the balloon changing 4 seconds after they start moving? Note: As a sanity check, make sure your answer does not contradict your answer to part (i).

So, in conclusion, I would say that the question you've provided would not be a viable calculus question on its own, but could reasonably appear as a "first step" in a viable calculus question that is being scaffolded.

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This question overall looks good, but I'll analyze it more in depth. Let's start at the second line.

At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon starts rising into the air at 12 ft/sec. Is the distance between the person and the balloon increasing or decreasing 4 seconds after they start moving?

On the surface, this looks like a good question. However, there is only one problem (Mentioned by @Sue VanHattum):

Calculus allows you to find the rate at which that distance is increasing. Logic (no calculus needed) tells you that it's increasing. The scenario is fine. The question (increasing or decreasing) makes no sense to me.

Now, to understand if this would be viable for a Calculus 1 course (I assume this is a question for a test early in the year), try looking at it yourself from the perspective of one of your Calc 1 students seeing the question on the test. If there's anything that you find that they might find confusing (i.e., they should already know that the question is stupid, since obviously, as mentioned previously, it's going to be increasing), then try to edit that part specifically so that it makes sense. In fact, I would probably rewrite the question as follows:

A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon starts rising into the air at 12 ft/sec. What is the rate at which the distance between the person and the balloon is going up?

I could probably write my rewrite of the question a bit better, although I'm not exactly sure how if that's the case.

So overall, if this is supposed to be a question on a test near the beginning of the year, then this is a good math question, however I would suggest wording the overall question a little better.

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    $\begingroup$ "what is the rate ... going up" is ambiguous. The person may think you're asking only about the y-axis component of the rate. $\endgroup$
    – shoover
    Oct 18, 2023 at 15:42
  • $\begingroup$ @shoover ah my bad. I know that definitely could have been worded way better, just am unsure how to. $\endgroup$
    – CrSb0001
    Oct 18, 2023 at 15:44
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    $\begingroup$ "increasing" is fine $\endgroup$
    – shoover
    Oct 18, 2023 at 15:50
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    $\begingroup$ I know that definitely could have been worded way better, just am unsure how to. --- What is the rate of increase in the distance between the person and the balloon? $\endgroup$ Oct 18, 2023 at 15:52

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