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## Algebra (all content)

### Course: Algebra (all content)>Unit 7

Lesson 13: Average rate of change word problems

# 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.

## Want to join the conversation?

• Isn't the first answer choice in this video the simplified version of the correct answer?
• No, because W(25) represents the amount of water remain after 25 seconds had passed, while W(0) represents the amount of water from the beginning (full amount).
• i finished functions and everything is marked in blue but it says that i did only 62/67 in functions
• 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.
• 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?
• 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.
• 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?
• 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).
• 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?
• 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.
(1 vote)
• No, they are not the same. W(0) in this problem does not mean "W times 0". It is function notation for "find the value of function W when t = 0).

Hope this helps.
(1 vote)
• Shouldn't amount of remaining water remaining W(25) be subtracted FROM initial water, W(0)?
• No. Always read and reread word problems carefully. The problem tells you that the function W(t) calculates the amount that remains after "t" seconds.
Hope this helps.
• 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)
• 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)
• 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)
• 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)

## 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.