A boat is moving towards the beach line at metres/minute. On the boat is a rotating light, revolving at revolutions/minute clockwise, as observed from the beach. There is a long straight wall on the beach line, as the boat approaches the beach, the light moves along the wall. Let equal the displacement of the light from the point on the wall, which faces the boat directly. See the diagram below. Determine the velocity, in metres/minute, of the light when metres, and the distance of the boat from the beach is metres.
Mathematics Specialist Semester 2 Exam 2018
The light is rotating at revolutions/minute, which means
We want to find and we know and .
We need to find an equation connecting and .
Differentiate (implicitly) with respect to time.
Now we know , and , using pythagoras we can calulate the hypotenuse.
I have been reading An Imaginary Tale – The Story of by Paul J Nahin, which is fabulous. There was a bit in chapter 4 where he found the closed form of the generalised Fibonacci sequence. I thought it would be a good exercise to find the closed from of the Fibonacci sequence.
Just to remind you, the Fibonacci sequence is
and it is defined recursively
That is, the next term is the sum of the two previous terms, i.e.
Now the starting off point is slightly dodgy as it involves and educated guess as Paul Nahin writes,
How do I know that works? Because I have seen it before, that’s how! […] There is nothing dishonourable about guessing correct solutions – indeed, great mathematicians and scientists, are invariable great guessers – just as long as eventually the guess is verified to work. The next time you encounter a recurrence formula, you can guess the answer too because then you will have already seen how it works.
We start with
This means is
or
Hence and we can use the initial conditions and to find and