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

MATH110 December 2015

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Question 05 (d)
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. (d) Is the ball speeding up or down at $t={\frac {\pi }{6}}$ ? Explain.

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?

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Hint
Recall that a function is increasing/decreasing at a point if its derivative is positive/negative at that point.

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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. We need to determine whether the speed function $v(t)$ is increasing or decreasing at $t={\frac {\pi }{6}}$. This means that whether $v'(t)$ is positive or negative at $t={\frac {\pi }{6}}$. We note that $v'(t)$ reperesents the accelation which is in fact the second derivative of the position function $h(t)$, so $h(t)=5+0.1\sin(2t)\Rightarrow h'(t)=v(t)=0.2\cos(2t)\Rightarrow h''(t)=v'(t)=a(t)=-0.4\sin(2t)$ which gives $a\left({\frac {\pi }{6}}\right)=-0.4\sin \left({\frac {\pi }{3}}\right)=-0.4\cdot {\frac {\sqrt {3}}{2}}<0\Rightarrow \$ The ball is speeding down at $t={\frac {\pi }{6}}$.