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water flows out of a storage tank at a rate defined by the function v prime of t as shown above t is in minutes v prime represents the volume flow rate of water in gallons per minute what is the rate at t equals 12 minutes now to find the rate we could simply just plug it in we need to find v prime of 12. so all we got to do is replace t with 12. so its going to be 100 plus 2.43 times 12 minus 0.05 times 12 squared so im going to do this one step at a time 2.43 times 12 is about 29.16 12 squared times .05 is 7.2 so if we add these values the rate at 12 minutes is going to be 121.96 liters per minute actually not liters but gallons per minute so thats all you have to do for part a now what about part b how can we calculate the number of gallons of water that will flow out of the tank in the first 30 minutes how can we do that well theres something called the net change term the integration of f prime of t which represents the rate of change of f with respect to time this is simply e
