Average rate of change word problems
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Karina drained a bucket of water. Let W(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 25 seconds, the amount of water remaining decreased by an average of four milliliters each second. 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 rate of 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 is equal to a decrease. The water decreased by four milliliters each second. So if we're decreasing, if our 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 4 milliliters, I'll just write that as mL, negative 4 milliliters per second. Now can we write this in a more "mathy" way? "The average rate of change of W over the first twenty five seconds?" The rate of change of W is going to be our change in W over our change in time. It's our change in W over the first 25 seconds, divided by the change of time over the first 25 seconds which is just 25 seconds. Our change in W is going to be our finishing amount remaining in the bucket, so W(25). That's how much we have at the end of this interval that we care about, how much water is remaining, minus how much water we started off with, divided by how much time goes by. And we could say "Hey, you know, we finished it at the 25th second, we started at the 0th second, or 25 minus 0 is just going to be 25." This expression I just wrote is the average rate of change of W over the first 25 seconds. Notice the way I wrote it. 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 units up here in the numerator, this would be in milliliters, and down here would be in seconds. So it makes sense that this would end up being in milliliters per second. But anyway, which of these choices have that? I have one more choice down here. This one over here looks exactly like what I just wrote. Now, a tempting one might be this one up here and the only difference between this one and this one is that we have a positive 4 over here. But keep in mind what this would imply. If W(25) minus W(0), in order for this to be positive (because we're dividing by a positive 25), 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 than you do initally. So that minus that needs to be a negative value. If you have a negative value up here, and you divide by a positive value, you should get a negative value. 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.