Science:Math Exam Resources/Courses/MATH101 B/April 2025/Question 07 (b)

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MATH101 B April 2025

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  •  April 2024  •  

Question 07 (b)

A small rocket propels itself straight upwards. From $t=0$ to $t=4$ minutes, its velocity is given by: $$v(t)=\frac{100}{1+t^3}\text{meters/min}.$$ When we were calculating the error of our approximation, we calculated that $$|v^{″}(t)|=\left|600t\frac{1-2t^3}{(1+t^3{)}^3}\right|\le 307200(=64\cdot 4800),$$ for all $0\le t\le 4$, which is too much. But we are dedicated and want to make our estimation work by using more trapezoids. To make our error at most 64 meters, using the bound given above, how many trapezoids would we need? Simplify your answer.

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
Begin by writing out the error formula and fill in everything you know. Your only remaining variable should be the number of trapezoids $n$. You want the error to be less than $64$, so ask yourself, what condition does $n$ need to satisfy in order to make the inequality hold?

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
To calculate the error, we will use $M=64\cdot 4800$ and hence, we want $\text{Error}\le \frac{M}{12}\frac{(b-a{)}^3}{n^2}\le \frac{64\cdot 4800}{12}\frac{4^3}{n^2}\le 64\cdot 400\frac{64}{n^2}\le 64.$ This means we need $n^2\ge 64\cdot 400$, so $n\ge 8\cdot 20=160$. Therefore, we can take $n=160$.

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