1) A ball is falling through the air and experiences air resistance. It’s velocity in metres per second at time t is given by dx/dt=250(e^(-0.2t) -1) where x is the height above the ground.

What is it’s eventual speed?

2) A tap on a large tank is gradually turned off so as not to exact any hydraulic shock. As a consequence, the flow rate while the tap is being turned off is given by dV/dt= -2+0.1tm^3/s.

I worked out the function to be V=-2t+1/20t^2+520.

The question is suppose that it is necessary to let out a total of 300m^3 from the tank. How long should the tap be left fully on before gradually turning it off?

3) over spring and summer the snow and ice on a mountain is melting at a rate of dI/dt = -5+4cos Pi/12 t where t is the time in days and I is the tonnage of ice.
A) Explain from the given rate why this ice is always melting?
B) The beginning of the next snow season is expected to be four months away (120 days) show that there will still be snow left on the mountain then. I worked out The function to be I= 18 000 -5t+48/pi sin Pi/12t

Q1 Eventual speed means speed as time approaches infinity, e^(-0.2t) will approach zero as t approaches infinity, therefore v=250(0-1) which is -250ms-1

Q2 At dV/dt at t=0 is the flow rate if the tap is left on fully which will be -2, then let dV/dt=0 such that you find the time it takes for the water to be "gradually turned off" which is 20s, subbing this into the V equation which did not have many brackets so I couldn't do it, you would get the amount of water lost which I will call W. Then 300=W+(2)T. Then T would be (300-W)/2.

I'm slightly concerned about the V function you worked out though.

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