Science:Math Exam Resources/Courses/MATH102/December 2017/Question 15 (a)

MATH102 December 2017

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Question 15 (a)
A grasshopper caught in a spiderweb rhythmically kicks its legs, causing the web to bounce up and down. The height of the grasshopper (in cm) after ${\textstyle t}$ seconds is $$h(t)=3+3\sin \left({\frac {5}{3}}t\right)$$ centimetres. (a) What is the period of this function?

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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
The period of the sine function is ${\textstyle 2\pi }$. In other words, for any $x$, $\sin(x+2\pi )=\sin x$.

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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. By the Hint, using the periodicity of the sin function, we have $$\sin \left({\dfrac {5}{3}}t+2\pi \right)=\sin \left({\dfrac {5}{3}}t\right).$$ Factoring ${\textstyle {\dfrac {5}{3}}}$, we have $$\sin \left({\dfrac {5}{3}}\left(t+{\dfrac {6\pi }{5}}\right)\right)=\sin \left({\dfrac {5}{3}}t\right).$$ This implies that the period of $\sin \left({\dfrac {5}{3}}t\right)$ is ${\dfrac {6\pi }{5}}$. Since a constant multiple of a function (a vertical dilation of a graph) and a translation don't change the period of a function, the period of $h$ is same with that of $\sin \left({\dfrac {5}{3}}t\right)$. More precisely, let us multiply by ${\textstyle 3}$ and add ${\textstyle 3}$ to both sides: $$3+3\sin \left({\dfrac {5}{3}}\left(t+{\dfrac {6\pi }{5}}\right)\right)=3+3\sin \left({\dfrac {5}{3}}t\right).$$ This means that $$h\left(t+{\dfrac {6\pi }{5}}\right)=h(t).$$ So the period of ${\textstyle h}$ is ${\textstyle {\dfrac {6\pi }{5}}}$. Answer: the period is ${\textstyle {\color {blue}{\dfrac {6\pi }{5}}}}$.