2000 q10 part B (1 Viewer)

cakes

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can someone explain to me how to do part (i) ? wtf is k/t and what the hell is t > T?

i have noo idea what its talking bout!! grrr :(
 

Wahlito

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I just tried that question today too and i am absolutely stumped. Even the answer does not make sence to me
 

tempco

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I remember doing this question... if anyone can post it up, I'm sure I can help.
 

cakes

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The first snow of the season begins to fall during the night. The depth of the snow, "h", increases at constant rate through thte night and the following day.
At 6am a snow plough begins toc lear the raod of snow. The speed, v km/h of the snow plough is inversely proportional to the depth of the snow.
(This means v= A/h where A is a constant :confused: )
Let x km be the distance the snow plough has cleared and let t be the time in hours from the beginning of the snowfall.
Let t=T correspond to 6am.

i) Explain carefully why, for t > T, dx/dt = k/t where k is a constant.

ii) In the period from 6am to 8am the snow plough clears 1km of road but it takes a further 3.5 hours to clear the next kilometre.
At what time did it begin snowing?
 

cakes

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I actually understand the solutions to part i now.. but if i got this in an exam, there'd be no way i could figure it out
 

ballerinabarbie

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ok i'll try

i) the speed of the snow plough is v= dx/dt
and the rate=dh/dt
v=A/h, and dx/dt=A/h

The depth of the snow is h=bt (where b is a constant rate)

therefore, dx/dt=A/bt=A/b/t
dx/dt=k/t

ii) At t=T+2, 1km of road is cleared
therefore, 1=(integral from T to T+2)k/t dt
1=[klnt](from T to T+2)
1=k[ln(T+2) - lnT]
1=k ln(T+2/T)

we use 1 again, because 1km of road is cleared in the next 3.5hrs
1=(integral from T+2 to T+5.5)k/t dt
1=[kln T+5.5/T+2] (because loga/b = log a - log b)

we make this equal to k ln(T+2/T) because they are both equal to 1

so k ln (T+5.5/T+2) = k ln (T+2/T)
cancel the k and the ln

T+5.5/T+2 = T+2/T (cross multiply)
T^2 + 5.5T = T^2 + 4T + 4

1.5T = 4
t= 4/1.5 = 2hr 40min

therefore it starts snowing at 3.20am

hope that helped... its off my maths tutors notes!! the first part is probably not explained very well... but i hope the second part makes sense
 

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