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Average rate of change word problem: equation

Average rate of change tells us how much the function changed per a single time unit, over a specific interval. It has many real-world applications. In this video, we represent the average rate of water draining with an algebraic expression.

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  • leafers seed style avatar for user tommyyankeefan
    Isn't the first answer choice in this video the simplified version of the correct answer?
    (10 votes)
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    • winston baby style avatar for user Michellesurreptitiosly
      To expand on what Vu said, what this equation is asking is to find the average rate of change of the amount of water left in the bucket after t seconds. To find the average rate of change, we do the change in y / change in x. Change of y would be our y-ending point (how much water is left at the end), W(25) and then subtract our y-starting point (how much water we have at the beginning, W(0).
      With our change in x, it would be our x-ending point (how much time has passed), 25 and then subtract it from our x-starting point (the time when we first start draining the bucket), 0 which is shortened to change in x = 25
      (3 votes)
  • male robot donald style avatar for user Yeshin Yoon
    i finished functions and everything is marked in blue but it says that i did only 62/67 in functions
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      The best way I know to figure this out is to go to the mission dashboard (likely same screen where you found 62/67). The circle that show amount mastered usually has a link under it that says "show all details". If you click this, you see squares for each skill and their associated color. Look for the ones that are not dark blue. Position your cursor on one and it will tell you the exercise set you still need to master.
      (7 votes)
  • blobby green style avatar for user mcruz
    How do you know W(25)? How would we know how much water is remaining, our finishing remaining amount in the bucket is 25, how?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      The 25 is not the amount of water in the bucket.
      The 25 is Time, specifically 25 seconds and is defined in the problem. Time (seconds) is also the Input value to the function W(t). W(25) = the output of the function after 25 seconds. We don't know its actual value because we weren't give the actual function. But, Sal is showing that you can write an expression for the rate of change (slope) by using function notation to represent the output value at t=25: W(25) and t=0 : W(0).
      Hope this helps.
      (3 votes)
  • duskpin ultimate style avatar for user sageR221b
    At he says that 25 is the remaining water in the bucket and 0 is the starting amount. How does that work when the amount of water is decreasing?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      w(25) is not the same as 25, and w(0) is not the same as zero. W(25) is read as w of 25, so it means the amount of water in the bucket at time 25 sec, and w(0) means the amount of water in the bucket at time 0 (which is what was in the bucket at the start).
      (3 votes)
  • cacteye green style avatar for user Ricardo L&SR
    This video contradicts the video before. If the amount is decreasing, the answer should be a negative number (like here -4), not positive as we saw on the video before (55m/s, instead of -55m/s). Am I right?
    (0 votes)
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    • hopper happy style avatar for user Sheng
      Well, in the last video, the question asked how much her height decreased at some rate and it decreased by positive since decreasing by a negative is a double negative.

      In this video, it asked for the equation that fitted the statement. So if the statement said that the average rate of change was decreasing, than the average rate of change is negative.
      (10 votes)
  • male robot donald style avatar for user andresHernandez803
    Answer d is the same as answer a.
    (1 vote)
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  • leafers ultimate style avatar for user Eddie Ryuzaki
    Shouldn't amount of remaining water remaining W(25) be subtracted FROM initial water, W(0)?
    (0 votes)
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  • blobby green style avatar for user zack rivers
    I think the question is constrained a little bit because of English but I think the answer to the English question should be positive. The numerical answer is -4 (in English this means decreasing by 4) but the English answer should be "four" because the English answer is "the water is decreasing by four milliliters per second" not "water is decreasing by negative four milliliters per second". A little pedantic, but I just thought it should be said.
    (1 vote)
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  • blobby green style avatar for user gna0905
    From to , Sal explains something, and I don't understand what he is saying/what he is trying to convey. I would appreciate if someone could explain! Thanks!
    (1 vote)
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    • piceratops ultimate style avatar for user Yuri
      He says, that if this equation is true:
      (W(25)-W(0))/25 = 4
      and 4>0, then the left side is positive:
      W(25)-W(0)>0
      W(25)>W(0)
      and we get that after 25 seconds we have more water, and the water tank is filling with
      water, that is not true.
      (1 vote)
  • blobby green style avatar for user gna0905
    Starting at , I really started to get confused about the whole equation that Sal began to write. Can someone explain? I would greatly appreciate it.
    (1 vote)
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Video transcript

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.