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CCSS.Math:

the cozy car company ships some of their new cars to Japan and Vietnam the number of cars that will be shipped to Japan during the next e months is modeled by the function J of T is equal to 2 to the teeth power the number of cars that will be shipped to Vietnam during the next t months is modeled by the function V of T is equal to 2t squared which country will which country had received more cars from the cosy car company after 5 months or will have received after 5 months so let's see how much Japan is going to receive after 5 months so T is in months so j5 is going to be equal to 2 to the fifth power which is equal to 2 times 2 times 2 times 2 times 2 and let's see 2 times just 4 8 16 32 so Japan will have received 32 cars and Vietnam so V of 5 is going to be 2 times 5 squared which is going to be 2 times 25 which is equal to 50 so based on these two models for how much they're going to receive after T months after 5 months Vietnam is going to receive Vietnam is going to receive more cars so I guess the answer to that is Vietnam Viet Nam will have received more cars after 5 months which country had received more cars from the cose car company or will have received more cars after 7 months so once again let's try this out J of 7 is equal to 2 to the seventh power let's see 2 to the 6 is going to be 32 times this is we can view this as 2 to the fifth times 2 times 2 which is going to be equal to this is going to be equal to 32 times 4 which is 128 cars after 7 months will have gone to Japan and to Vietnam V of 7 is going to be equal to 2 times 7 squared so that's equal to 2 times 49 which is equal to 98 cars so after seven months Japan would have received more cars so Japan Japan will have received more cars after after seven months and so this is interesting we see that the exponential function notice where you have the T as the exponent that although it might start off a little bit slower than this what's essentially a quadratic function when do you have something squared it starts off slow after five months you would have shipped fewer cars than using the quadratic model right over here but then it more than catches up and it starts to increase it at a faster and faster rate and even by seven months its able to pass up the quadratic function which country will receive will have received more cars from the cose car company f will the country which received more cars from the cose car company after seven months continue to receive more cars than the other country in future months yeah absolutely once the exponential function passes up the quadratic function it just goes faster it just keeps increasing at a faster and faster rate and you could see that if we wanted to do if we wanted to compare eight months so J of eight this is the exponential function this would be two to the eighth power which would be this times two they'd get two hundred and fifty six cars and V of eight V of eight is going to be 2 times 8 squared which is 2 times 64 which is 128 so notice now now that we would have shipped twice as much to Japan as Vietnam which isn't what the case right over here we ship more to Japan the Vietnam but not twice as much and we could keep going we could if you want you could go to jail you could go to jail J of 9 is 2 to the 9th power which is going to be 256 times 2 or 512 cars while V of 9 V of 9 is going to be 2 times 9 squared which is 2 times 81 which is 162 so now it's more way more than double actually more than triple so you see that once you get past that in those initial few months the the exponential function is increased a much much much faster rate and we could actually visualize that let's actually get out a graphing calculator to visualize these two things to see how that is happening so let's graph it so the first one let me graph the exponential so two to the two to the well I'll just say X power we'll say X is our independent variable here so two to the X power and then let's do the quadratic one this is y2 lwv of two let's say 2 times X 2 times x squared now let me set the range so let me set the range here so let's see let's say X starts at zero and then let's say it goes up to let's say it goes up to 10 let's say it goes up to 10 the X scale could be 1 or X scale to be 1 now wise minimum let's say list will start at 0 and then Y max well let's say let's go to a thousand let's go to a thousand let's make the Y scale one hundred one hundred and now I think we're ready to graph so let's graph this let's see what happens so it's munching on things so that right over there that's the exponential and then there you see right over there you have the quadratic and actually let me zoom in a little bit on this so that we can see where they pass where I'll actually let me let me actually let me zoom in a little bit on this so I'll do with the box so that we can really see wicket we can see where they are attempt to see where they pass each other up so I'm going to start there I'm going to make my box let's see go whoops so we are using a calculator on a computer like this but you see you definitely see even what we've already graphed that the exponential really just starts to shoot up while the quadratic is just going well it's still increasing at a decent pace but nowhere near as fast and the difference is becoming more and more pronounced as time increases so let me just make sure that's as low as that and let's let's graph over in this range right over here so let's see that right over there that's the exponential function that's 2 to the T power and then that right over there is the quadratic so you see you definitely see actually let me you definitely see that earlier on the quadratic has higher values and you see that right over here after five months we ship more cars with to Vietnam but then the exponential passes it up and then just keep shooting faster at an ever-increasing pace