So, according to data from the American Hospital Association, the rate of change in the number of hospital outpatient visits and millions in the United States each year from 1982 the president can be approximated by F Prime of T is equal to 0.1483 tee times T minus 1982 the 0.75 power. So we want to use the fact that in 1980 there was 262,951,000 outpatient visits, and we want to find a formula that will approximate number of outpatient visits as a function of time. So we're told that this is a rate of change in the number of hospital out patient visits. So remember rates of change are really derivatives. So this F prime of T if we integrated won't get a function for hospital outpatient visits. So we'll have f of t equal to the integral primer t. So I'm first going to go ahead and pull out this constant here. What I meant to raise hopes one colour. So 0.1483 integrate two T minus 1980 to the 0.75 power with respect to teach. Now, looking at this, you might see that if we let if we do a substitution of T minus 1980 we can go ahead and solve this. So let's just go ahead and off on the side, right? All what that would be. So if we have u is equal to t minus 1980 that will tell me that you Earth d'you by DT will just be one. Or do you is equal to d t? Oh, I just want the same thing again. The U is equal to duty and something else will need to know his we need to solve for what t is because we'll end up that this tea right here won't get cancelled out when we make this substitution. So we'll go ahead and also solve for t and we get t is equal to you. Plus 19 baby. Really? I got a little bit better So t as you plus 19. 80. So we use these two facts and I'm doing that will end up with 0001483 into the role of So I plug in what tears will be you plus 1980 times. So this will be to the 0.5 power and then you hear and d t is just equal to d you. So we have that. Now we can go ahead and distribute you to the 0.75 power, and we can go ahead and apply power rule twice. So you becomes 1.75 and then we'll get 1982 the view 0.75 Do you now we can go ahead and integrate over these by using power rule. So you get 0.21483 So this will be you to the 2.75 over our new power. Two points five was 1980 hoover. For the first you do 1.75 power divided by its new power, plus some constant seats. All right, so what I'm gonna first do before I plug you back in is factor out some of these use. So a factor are you to the 1.75 power and you don't need to do this, but I just want to do it to clean this up a little bit before we start doing stuff. So they're 001 for a three. You to the 1.5. And then on the inside this will become you over 2.75 I mean, it's a little bit. So you over at 75 plus 19. 80/1 190.7, right? Plus C r. Elvis, go ahead and plug. Are you in and doing that? 10.1 for eight. Three. And they remember. Are you Is this right here? So we're going to plug that in the T minus 1980 and this will be to the 1.7 by power and then I need to plug in you up top here. This will be T minus 1930 over 2.75 was 19. 80/1 190.75 plus C. And so this here will be our general f a T. Now, what we want to do to solve for T is used the fact that in 1980 there were about this many patients. So sure, let me just use your mouth a little bit. So what we want to do is look at half of 1980. We know this is going to be 0.1483 Hello, princess. Right here, minus 1980. And then I'm just gonna put some dots right here. I don't have to rewrite it again, because if you notice when we plug in 1984 1984 1980 into this part 1980 minus 1980 is zero. And then this time, zero times that will just be zero. So you end up with half of 1980 is equal to see. And 1980 was 262 million. 951,000. So now we can use all this to rewrite the equation and end up with the following. Let me erase some of this right here. All right. Oh, so we have that of of T is equal to 0.14 Heat overheat. T minus 1982. The 1.75 power five TV by this 1980 over 2.75 plus 1980 over 1.75 and are interesting. Not a little bit plus our constant right here. Mm. Plug this in there. 262 91002 Yeah, So we have this big equation here for how many outpatients we will get. So we'll need this for the next part of the equation For the next part of the question which is use this to protect the number of out patient visits in 2020. So the equation that we have, let's go ahead and rewrite it again. So we have Earth of T. V is equal to 0.0 01483 T minus 1980 to the 175 power T minus 19 80/2 190.75 plus 19 80/1 190.75 plus our constant. 262 951 00 Yeah. And since we want to find in the year 2020 we'll plug in 2020 into here. So f of 2020. So well said if I plug in 20 into each of these, it will be with 40. So I'm just gonna go ahead and do that at one step. Zero point color. So 0.1483 to be 1.75 power with 40 on the inside since. Remember, I'm plugging 2020 n 14, minus 1980. We'll have so 40 talk divided by 275 plus 19 80/1 190.75 and then plus our constant to 6 to 9 ft one individual. So now 40 raised to the 1.75 power is about 336 and then multiplied by 0.1483 gives about 0.94 So now I'll start approximating. So this is approximately so. First, this here. Mm. This here is about 0.94 35 So all of this right here is going to be about so 40 divided by 2.75 is about 14 5 and then add that to 1980 divided by 1.75 So there will be about 1145 0.97 So I'll need to multiply these two things together and then also add to 16, 51 So, supplying those two together, we'll get a 20.9435 So about almost 1100. This is approximately where you Yeah. 10. 81 0.2 and then plus 2 16, +951000 and then adding these two together gives approximately 262 95 2081 0.2. So since we're talking about a number of patients, I'll just go ahead and take off this 0.2 and round it down. So in the year 2020 they will expect to have about this many patients or outpatient visits.