The amount of air in a balloon at any time t is given by V(t). Determine if the balloon is being filled with air at any moment of time t.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Here the amount of air in the balloon at any moment of time t is given as V(t). Now if the balloon is being filled or not can be determined by finding the value of the differential of the function V(t).

At any t if the balloon is being filled, the amount of air at t+ dt is going to be greater than the amount of air at t.

=> V(t+ dt) > V(t)

Now if dt --> 0, we get the instantaneous change in the volume of air at any time t.

V'(t)= lim dt--> 0 [{V(t+dt) - V(t)} / dt ]

So we need to find V'(t).

Substitute the value of time t at which you want to know if the balloon is being filled or not. If V'(t) > 0 , the balloon is being filled, if V'(t)< 0, the balloon is being emptied and if V'(t) = 0 it implies there is no change in the volume of air.

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