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Stressed, scared and stuck :/ - HSN forum

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Stressed, scared and stuck :/


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#1 trippingeyes

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Posted 19 May 2009 - 10:49 AM

I have a couple of questions that I'm stuck on, will anyone possibly help? (These are from Winter 2002 btw)
Optimization :/
7. A rectangular beam is to be cut from a cylindrical log of diameter 20cm.

The diagram (circle with rectangle inside touching edges) shows a cross-section of the log and beam where the beam has a breadth of w cm and a depth of d cm.

The strength S of the beam is given by

S=1.7w(400-w^2).

Find the dimensions of the beam for maximum strength.

Ans
d=20\/(2/3)
* The \/ is meant to be a square root :/


11. I'm really confused by how to include a restriction on the y-axis ??? Could I add in an equation for a line that is horizontal to the x-axis?

I hope this is clarified well enough, really stuck and terrified about Thurs

#2 Garden

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Posted 19 May 2009 - 12:06 PM

QUOTE (trippingeyes @ May 19 2009, 11:49 AM) <{POST_SNAPBACK}>
Optimization
7. A rectangular beam is to be cut from a cylindrical log of diameter 20cm.

The diagram (circle with rectangle inside touching edges) shows a cross-section of the log and beam where the beam has a breadth of w cm and a depth of d cm.

The strength S of the beam is given by

S=1.7w(400-w^2).

Find the dimensions of the beam for maximum strength.


Don't panic, for the love of Peter - optimisation is not necessary to pass and usually only about 4 marks which is not enough to fail so you CAN skip this if you're only looking for a C.

Ok, so we try it.

S(x) = 1.7w(400 - w^2)
= 680w - 1.7w^3

S'(x) = 680 - 5.1w^2

Maximum turning point when S'(x) = 0
Thus 680 - 5.1w^2 = 0
5.1w^2 = 680
w^2 = 133 1/3
w^2 = 400/3
w = 20/root3 = 60/3 = 20

Now, use a nature table (hard to type out but I'll try)

x | <- | 20 | -> |
S'(x) | +ve | 0 | -ve |
Slope | / | - | \ |

(You can find S'(x) for a value lower and higher than 20 or you can assume it's right)

Then, because we know that d = 20 from the question:

w = 20 and d = 20 will give the maximum strength.

Hope this helps!
Beware negatives,
for they will haunt you in haikus
and in exam rooms.





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