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Mathematics Question Thread
RuiAce:
Assumed we wanted to minimise time:
P.S. Woah Jake what are you doing with the trig!
Happy Physics Land:
--- Quote from: amandali on February 27, 2016, 07:55:06 pm ---how do you do this ques thanks ;D
a man in a rowing boat is presently 6km from the nearest point A on the shore. He wants to reach as soon as possible a point B that is further 20km down the shore from A. If he can row at 8km/hr and run at 10km/hr , how far from A should he land? ans: 8km
--- End quote ---
Hey Amanda:
Another great question from you amanda! Ok, so when I first looked at the question, my instinct tells me I will have to draw a diagram, because it involves distance and speed and even worse theres also time. And I will have to admit, despite my distaste towards drawing diagrams, it is essential for you to draw one because graphics help you to visualise stuff. So next time when you see these types of questions, definitely draw a diagram, its worth the time.
Ok and then I manipulated pythagorus theorem, made an algebraic expression for the time to travel distance MS, made another algebraic expression for the time to travel distance BS. The time taken to travel the total distance now becomes T(MS) + T(BS), where T =time. So after establishing such a relationship, the rest is straight forward: simply repeat the conventional process of simplifying, differentiating, make derivative = 0, and then find stationary point. Afterwards you test a value on both sides of stationary point to determine its nature (I.e. maximum or minimum) and then make a concluding statement for your answer. Actually it is quite crucial to include a concluding statement, because that makes it clearer for the marker what your final result is and you would less likely to be deducted a mark on not stating the result clearly.
Anyways, my solution as below:
Sorry for the messy working btw, if you have any further questions dont hesitate to ask! :)
Best Regards
Happy Physics Land
jakesilove:
--- Quote from: RuiAce on February 27, 2016, 09:20:05 pm ---Assumed we wanted to minimise time:
(Image removed from quote.)
P.S. Woah Jake what are you doing with the trig!
--- End quote ---
So, this is a wayyyy easier solution. There are so many ways to answer questions like these, so whichever makes more sense to you! Still, thanks everyone for participating and answering questions on the forums!
My method is definitely more tricky, because I have a tendency to over-use trig. If you can use algebra, that's usually a better and simpler solution.
Jake
16ebond:
Hey Jake
So we have been doing e in class and I am a bit stuck on how to answer this question.
I have attached the question, so hopefully the file will open.
Thanks heaps
Em :)
RuiAce:
I'm not sure if this was meant to be an unanswered question but here's a solution
Pick an arbitrary point (a,0) to be the first vertex of the rectangle. By default (a, e^(-a^2)) is also on the rectangle. But because it HAS TO BE A RECTANGLE and we have an even function
(-a,0) and (-a, e^(-a^2)) must lie on the rectangle. This is most easily shown with a diagram.
So the breadth and length of the triangle are:
Hence we can combine these to give an expression for area:
Set dA/dx=0 to maximise:
Reject x=0 though, because if x=0 we have no rectangle (we can't have a width of 0).
So just show that the remaining value gives a maxima using a table of values or the second derivative. Note that because the curve is an even function, the negative term can be ignored.
I would've used latex but I'm thoroughly lazy to on a phone (Hello, Jamon here, I went through and added LaTex ;D)
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