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End behavior of algebraic models

CCSS.Math: ,

Video transcript

a barista poured a cup of coffee the initial temperature of the coffee was 90 degrees Celsius at time t increased as as time t increased the temperature c of the coffee began to decrease exponentially and approached room temperature of 20 degrees celsius which of the following which of the following graphs could model this relationship so we're starting at 90 degrees Celsius it looks like all of the graphs start at 90 degrees Celsius at T equals zero and we are going to get we're going to approach the room temperature of 20 degrees Celsius so this first one does approach the room temperature of 20 degrees Celsius as T increases now this one and when T is 70 and when T is 70 I'm assuming this is in minutes when T is 70 it looks like it has the temperature going to zero degrees Celsius so that that cup of coffee is gonna start freezing so I think I could rule out B also this looks like a linear model not an exponential one C it does get us to this end state that stays at 20 degrees but it doesn't look like an exponential model looks like it's linearly decreasing and then it stops linearly decreasing after 50 minutes and then it just stays constant at that temperature of 20 degrees so even though it gets us to the right place it does not look like an exponential decay so I would rule choice C out as well so a is looking good D we are starting at 90 it does look like an exponential and exponential function we have exponential decay right over here we and we are approaching something but it's not the room temperature of 20 degrees Celsius we're approaching 30 degrees Celsius here so I'd also rule out D so a is looking good it's an exponential it's decreasing exponentially starting at 90 degrees Celsius and it's approaching the room temperature of 20 degrees Celsius let's do another one of these so it says let me scroll up a little bit so it says after after the closing of the mils the population of the town starts decreasing exponentially the graph below presents represents the population P in thousands of the town two years after the closing of the mill all right so it looks like the population starts at 40,000 its decreasing exponentially it looks like over time the population is approaching 20,000 people so what are the question here based on the graph with the mill closed what is the population of the town approached as time increases well we just said it as time increases it looks like it's coming close to its approaching 20,000 its approaching 20,000 it's already gotten below twenty two thousand four as you know looks like by two after 20 or 22 years we've already gotten below 22,000 so we're definitely below 30 or 40,000 but where we haven't gotten below 20,000 but we are approaching it we could even check our answer if we like