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Average rate of change word problem: table


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let T of T so capital T of lowercase T denote the temperature capital T in Windhoek Namibia measured in degrees Celsius when it's T lowercase T hours after midnight on a given day the table below lists a few selected values of the function alright when six hours after midnight our temperature is 19 degrees Celsius nine hours after midnight or nine a.m. 25 degrees celcius 13 hours after midnight which is the same thing as 1 p.m. our temperature is 31 degrees Celsius when did the temperature increase faster so this is between 6 & 9 a.m. so 6 & 9 a.m. 9 a.m. and 1 p.m. so that's between these two points or did the temperature increase at the same rate over both intervals well if we want to figure out the rate of increase the rate of increase so let's see rate of temperature change temperature change well this is going to be equal to our change in temperature capital T over our change in time and this triangle that's the Greek letter Delta it represents change in so change in temperature over change in time so let's think about how this what this is what this rate of temperature changes between these different intervals so between 6:00 a.m. and 9:00 a.m. so let me really in temperature over change in time so what was our change in temperature our temperature went up by 6 degrees Celsius 6 degrees Celsius let me just write it here 6 degrees Celsius and what is our change in time our time went up by 3 hours plus 3 hours so you're going to have a change of 6 degrees Celsius positive change of 60 degree Celsius over a positive change we've gone 3 hours into the future so for over 3 hours we increased our temperature by 6 degrees or you could say it's an average rate of change of 6 divided by 3 is 2 degrees Celsius per hour notice I'm just keeping the unit's the same degrees Celsius per hour so that's the rate of change between 6 a.m. and 9:00 a.m. now let's ask ourselves the same question between 9:00 a.m. and 1:00 p.m. so our change in temperature over change in time what is our change in temperature well our temperature goes up from 25 to 31 degrees Celsius that's another 6 degrees Celsius so that's another 6 degrees Celsius and but it's no longer 3 hours to go from nine hours after minute to 13 hours after midnight we're now doing it over 4 hours so plus 4 hours so 6 degrees Celsius over 4 hours and we actually don't even have to calculate you see that you've had you've had the same change but you've had to do it over more hours so this is a lower rate of change the temperature is increasing slower here it took it four hours to increase 6 degrees Celsius well over here it took it only 3 hours so immediately you might recognize that look this is going to be faster if you increase 6 degrees Celsius over 3 hours that's faster than increasing 6 degrees Celsius over 4 hours but just to make the comparison a little bit just to make that compares it a little bit clearer let's actually just do the math here 6 divided by 4 well that's going to be 1.5 1.5 and degrees Celsius per hour and here you can make the direct comparison to degrees Celsius per hour is faster than 1.5 degrees celsius per hour so that's why we liked we liked this choice this choice up here