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# Solving equations by graphing: word problems

CCSS.Math:

## Video transcript

we're told to study the growth of bacteria a scientist measures the area in square millimeters occupied by a sample population the growth of the population can be modeled by F of T is equal to 24 times e to the 0.4 times T power where T is a number of hours since the experiment began here's the graph of F so I'm guess is going to be the output of this function is going to be the number of square millimeters after T hours alright so here we have the graph we see how as time goes on the square millimeters of our little bacterial population keeps growing and it clearly is growing or it looks like it's growing exponentially in fact we know it's going exponential because it's an exponential function right over here and they say when does the population first occupy an area of 400 square millimeters so pause this video and try to figure that out all right and this is a screenshot from the Khan Academy exercise so we want to say when does the population first occupy an area of 400 square millimeters let's see 400 square millimeters is right over there and so it looks like after seven hours that we are going to be 400 square millimeters or larger so it first hits it after seven hours so seven hours just like that now let's do the next few examples that build on this so if I go back up to the top and now we're told the same thing we're using square millimeters square millimeters to study the growth this is the function but then they add this next line here is the graph here is the graph of F and the graph of the line y equals 600 so they added that graph there and then they say which statement represents the meaning of the intersection point of the graphs all right so let's look at the choices here so and it says choose all that apply so pause this video and see if you can answer that all right so choice a says it describes the time when the population occupies 600 square millimeters so which statement represents the meaning of the intersection point of the graph so they're talking about they're talking about this point right over there so does that describe the time when the population occupies 600 square millimeters so that is the time when the population has indeed reached 600 square millimeters because that's the line y is equal to 600 square millimeters so I like that choice I will select it the next choice it gives the solution to the equation 24 times e to the 0.40 is equal to 600 well if you think about it this right over here in blue we've already talked about it that is y is equal to 24 times e to the 0.40 power this is y is equal to 600 so the T value at which these two graphs equal that means that they're both equal to the same y value or another way to think about it it means that that is equal to that or that 24 times e to the 0.40 power is indeed equal to 600 so I like this too it gives a T value where this is true so that's the solution to that equation it describes the situation where the air where the area the population occupies is equal to the number of hours that's definitely not the case because the area here is 600 square millimeters the hours looks like it's a little bit after 8 so they're definitely not equal it describes the area the population occupies after 600 hours no we don't have to look up there this t axis doesn't even go to 600 hours so we wouldn't select that as well now let's keep building and go to the next part of this and it says it says so once again we measure the area square millimeters to figure out the growth of the population the of us here we have two populations here it says the growth of population a can be modeled by F of T is equal to that we've seen that already but now they are introducing another population the growth of population B can be modeled by G of T is equal to this where T is the number of hours since the experiment began here are the graphs of F and G so now we have two populations they're both growing exponentially but at different rates and then it says when do the populations occupy the same area and says round your answer to the nearest integer and you could pause this video and try to think about that if you like well you can see very clearly that it looks like or they intersect right around there so that's the point in which they're going to occupy the same area it looks like it's about a hundred and seventy five square millimeters but they're not asking about the area they're saying when does it happen and it looks like it happens after about five hours so round to the nearest integer let's say five hours now let's do the last part so it's the same set up but now they are asking us a different question they are asking us which statements represent the meaning of the intersection points of the graphs all right so choice a it says and then pause the video again and try to answer these on your own all right choice a says it means that the populations both occupied about a hundred eighty square millimeters at the same time so let's see this that looks about right I had estimated 175 but you could call that 180 and it looks like that does roughly happen at around the fifth hour so it looks like they're occupying the same area at around the same time so I like that choice it means that at the beginning population a was larger than population B well the point of intersection is doesn't tell us what population was larger to begin with we could try to answer it by looking over here when time T equals zero when time T equals zero population a is the blue curve it is F and so it does look like population a was larger than population B at time equals zero at the beginning but that's not what the point of intersection tells us so they're not just asking us for true statements they're saying which statements represent the meaning the meaning of the intersection point of the graphs but that doesn't tell us about what the starting situation was it gives a solution to the equation 24 times e to the 0.40 is equal to 9 times e to the 0.60 well we already talked about that in the in the last example we really had one curve and that actually is the case because Y is equal to 24 times e to the zero point 4 T is the curve for population a and then y is equal to 9 times e to the zero point six T is the curve for population B and so the point at which these two curves intersect that's the point at which both this we're at a T value that gives the same so that this expression will give you the same y value as this expression or another way to say it is we're at the T value where this is equal to this so it does indeed give us a solution to the equation and then the last choice is it gives the solution to the equation 24 times e to the 0.40 is equal to 0 no that would happen if you want to know when it's equal to 0 you would look at the curve y equals 0 I'll do that in a different color which is right over here and see where it intersects the function f which is equal to 24 times e to the 0.40 but that's not what this point of intersection represents so we definitely wouldn't pick that one either