Science:Math Exam Resources/Courses/MATH110/December 2015/Question 05 (e)

MATH110 December 2015

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Question 05 (e)

A ball is floating on the surface of the sea. The ball’s height $h$ (in m) above the sea floor at time $t$ (in s) is given by $h(t)=5+0.1\sin(2t).$ Answer the questions below. Evaluate all function values and provide justification for your claims. Include units. (e) Find the initial position of the ball and then determine what is the earliest time $t>0$ at which the ball is back in its initial position.

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
Note that $\sin(x)=0$ if and only if $x\in \{n\pi :n\in \mathbb {Z} \}=\{\cdots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\cdots \}$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since $h(0)=5+0.1\sin(2\cdot 0)=5$, the initial position of the ball is $5$. Now, we want to find the first positive real number $t$ at which $h(t)=h(0)=5$. Note that $h(t)=5+0.1\sin(2t)=5$ is equivalent with $0.1\sin(2t)=0$ and $\sin(2t)=0$. Therefore, our problem is reduced to find the first positive real number $t$ such that $\sin(2t)=0$. Using the Hint, $\sin(2t)=0$ if and only if $2t\in \{n\pi :n\in \mathbb {Z} \}=\{\cdots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\cdots \}$. This implies that $2t=\pi$, i.e., $t={\frac {\pi }{2}}$ is the earliest time that the ball is back in its initial position. To sum, the answers are $\color {blue}5(m),{\frac {\pi }{2}}(s)$.