Science:Math Exam Resources/Courses/MATH103/April 2009/Question 06 (a)

MATH103 April 2009

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Question 06 (a)
Seeds are dispersed from a tree in all directions uniformly. The density of seeds that lands on the ground at a distance r away is approximately described by the function $s(r)=50\sin \left({\frac {\pi r}{10}}\right){\text{ seeds/m}}^{2},\quad 0\leq r\leq 10{\text{ m}}$ At what distance from the tree are the seeds most densely distributed?

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
The seeds are most densely distributed at the point where the function that describes the density has a maximum.

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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 we look for the maximum of s, we set its derivative to zero. $0={\frac {d}{dr}}50\sin \left({\frac {\pi r}{10}}\right)=50{\frac {\pi }{10}}\cos \left({\frac {\pi r}{10}}\right),$ $\cos \left({\frac {\pi r}{10}}\right)=0,\$ Due to the restriction on r the only possible zero is when ${\frac {\pi r}{10}}={\frac {\pi }{2}}$ which is attained at $\displaystyle r=5$ Note that $\sin(\pi (5)/10)=\sin(\pi /2)=1$ gives indeed the maximum value for sine and hence for s.

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