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# Average rate of change word problem: equation

Transkrypcja filmu video (w języku angielskim)
we're gonna drained a bucket of water let W of T denote the amount of water W measured in milliliters that remained in the bucket after T seconds which equation best represents the following statement over the first twenty five seconds the amount of water remaining the amount of water remaining the amount of water remaining decreased by an average of four milliliters each second each seconds so let me let me let me let me rewrite this let me paraphrase this and then maybe we'll be able to think about the math a little bit they're saying that the average average rate of change change of the amount of remaining water so I could say the average rate of change of W with respect to time over first 25 seconds 25 seconds is equal to is equal to a decrease the water decreased by four milliliters each second so it's going to if we're decreasing if I if I rate of change if W is going down our rate of change is going to be negative every second that goes by W is going to go down by some amount so it's going to be negative four milliliters I'll just write that is ml negative four milliliters per second negative four milliliters per second now can we can rewrite this in a more mathy way the average rate of change of W over the first twenty five seconds well the rate of change of W is going to be our change in W over our change in time we're such ancient uh below over the first twenty five seconds divided by the change of time over the first twenty five seconds which is just twenty-five seconds so this is going to be so our change in W is going to be our it's going to be our our finishing amount remaining in the bucket so W of twenty five that's how much we have at the end of this interval that we care about how much water is remaining - how much we started off with - how much water we started off with / how much time goes by and we could say hey you know we finished it second at the 25th second we started at the zero at second or 25 - zero is just is just going to be 25 so this right over here with this expression I just wrote this is the average rate of change of W over the first 25 seconds notice the way I wrote it I had when I write it like this it might be a little bit clearer this is our ending W minus our initial W and this is our ending time minus our initial time this last part just simplifies to 25 and they tell us that this is going to be negative 4 milliliters per second this is going to be equal to negative 4 and the unit's up here in the numerator this would be in milliliters and down here would be in second so it makes sense that this would be end up being in milliliters per second but anyway which of these choices have that and I have one more choice down here well this one over here looks exactly like what I just what I just wrote now attempting one might be this one up here and the only difference between this this one and this one is that we have a positive 4 over here but keep in mind what this would imply if W if W of 25 - W of 0 in order for this to be positive because we're dividing by a positive 25 in order for this to be positive then this would have to be positive then this would have to be positive in order for this to be positive that would mean that we have more water remaining after 25 seconds than we do after 0 seconds because in order for this to be positive this one has to be larger which means that somehow the bucket is filling up with water not draining but we know that the water is decreasing by an average of 4 milliliters each second so if we're decreasing this value over here needs to be equal to negative you have to have a lower value after 25 seconds then you do initially so that - that needs to be a negative value and if you have a negative value up here and you divide by a positive value you should get you should get a negative value and it also makes conceptual sense the water is decreasing the rate of change of water with respect to time should be negative because the amount of water is decreasing