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Numerade Educator



Problem 49 Hard Difficulty

The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day.
(a) Compute and interpret the average rate of change from 1990 to 2005. What are the units?
(b) Estimate the instantaneous rate of change in 2000 by taking the average of two average rates of change. What are its units?


(a) 1169.6

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Video Transcript

So on this problem we're given this table Uh in years since 1985 and oil consumption in thousands of barrels of oil per day and were first asked to compute and interpret the average rate of change abbreviate A R. O C. Average rate of change from 1990 2005. Okay. The average rate of change will be Oil consumption in 2005 minus oil consumption in 1990 divided by 2005 -1990. Okay well let's see. 2005 years. 2500 1985- 2005. That's 20 years. So that's here 1990 would be five years after 1985. That would be here. All right. So that means that this is equal to 84 77 -66-533 divided by 2005 -1990. Remember these are coordinated? So the input year for the first entry in the numerator is the first year in the denominator. Okay, so that means I now have 84 77 -66-533 Is 17544 over 15. Right, This is barrels per day divided by years. and so when I divide that by 15, So that is 1000 169.6 um barrels per day per year. So there's the average rate of change between those two years. Next it says to estimate the instantaneous rate to change In the year 2000. So where does the year? 2000 show up on here? Well, 18, That's it. 15, isn't it? Okay. And so what does that mean? Well, that means we do the average rate of change going down and then the average real change going up and we average them. Okay. So, so the average rate of change going down, I'll say is consumption in 2000 minus consumption in 1995 kind of ugly. Let's clean that up 1995 divided by 2000 -1995. Mer. We're keeping these coordinated. Okay, So from our table 2000, the year 2000 is 76 7 84 1995 is 70,099, divided by five. Right? Because 2000 and -1995 is five. So this is 6685 divided by five, which is 1337. Okay. And then the average rate of change going up the table yeah, will be the consumption In 2005 The consumption in 2000 divided by 2005 -2000. Okay. Yeah, so that is The consumption in 2005 is 84.077 Minour the consumption in 2076, 7 84. Over five years. Right, 84 077 -76784. So that's 7,293 over five, which is 14 58.6. Okay, So then I average these two. So the instantaneous rate of change In the year 2000 is one half times 1337 plus 14 58.6. Okay. And so when I do that, I get 1000, barrels, uh, her day or? Yeah. So there is the instantaneous rate using The average of two average rate to change. And it has the units on it as well.