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Friday, June 24, 2011

Y10 Add Maths Related Rates Quiz

Johhny asked Mary, why did Sir say: "You cannot step twice into the same river." I looked it up, replied Mary. He was quoting Heraclitus of Ephesus. It means that it is not the same river because the water you stepped into has already moved downstream and other water with possibly a different pH, temperature and biological oxygen demand has replaced it. Actually, said Johnny I think it means more than that. It is not the same “you” either. You have had the new experience of stepping into the polluted river, the chocolate bar you have eaten has released phenyl ethyl amine into your blood stream and an enzyme in one of trillions of your cell nuclei has been busy repairing a piece of DNA which was damaged by a cosmic ray that had travelled 15 billion light years from a dying star at the edge of the known universe!

Somwhow you forgot to say I am also prettier than I was yesterday. What do you know about Parmenides?” asked Mary? “Is he the new student all the girls are crazy about?” replied Johnny. Of course not said Mary. He’s another Greek philosopher. In his poem, On Nature, he explains how reality is one and change is impossible. Our senses cannot perceive true reality. As we focus on different parts of the same one reality we just have the illusion that things have changed. For example, your iPod has all the songs stored in it at once, but you can only hear one chord at a time. In the same way, a superior being outside of time could perceive our wholes life at once.

In the debate between these philosophers from very different viewpoints, mathematics takes the middle ground. Change exists but behind change we a permanent form.
The area of a triangle (see diagram) is growing at 3 cm2 per second.

i.e., dA/dt = 3 and the triangle remains self-similar

i) Find the rate of increase of the base, when the base is exactly 12 cm and the height is exactly 6 cm. [3]      
Note that h = ½ b  (Not just at time a particular time t but for the duration of growth of the triangle)
A = ½ b h
A = ½ b ( ½ b)
A = ¼ b2
dA/db = ½ b
dA/dt = dA/db × db/dt
3 = ½ (12) × db/dt
db/dt = ½

ii) Find the rate of increase of the height, when the base is exactly 12 cm and the height is exactly 6 cm.    [2]
Note that b = 2h and proceed as before
A = ½ b h
A = ½ (2h) ( h)
A = ½ (2 )h2
A = h2
dA/dh = 2h
dA/dt = dA/dh × dh/dt
3 = 2 (6) × db/dt
db/dt = ¼

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