The population of the United States (in millions) by decade is...

Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Estimate the instantaneous rate of growth in 1985.

Estimate the instantaneous rate of growth in 1985.

Accepted Answer

Welcome back, everyone. In this problem, a scientist is studying the growth of a bacterial population in our lab culture. The experiment started at 8 a.m. and the table shows the population T always since the experiment started. The population size B of T as a function of time T is modeled by the curve below. Here we are shown a table of the values, and we are also given a graph with our curve. Estimate the instantaneous population growth rate at 4:30 p.m. Now, if we're going to estimate this growth rate, the first thing we need to do is to figure out where on our graph 30 p.m. 4:30 p.m. sorry, would be. Now, we're told that the experiment starts at 8 and the table shows the population 2 hours since the experiment started. So when T equals 1, it's 1 hour after 8 a.m. or 9 a.m. Likewise, T equals 2 is 10 a.m., 3 is 11 a.m., 4, 120 p.m., 51 p.m., 62 p.m., 73 p.m., and 4, sorry, 8, 40 p.m. So 4:30, OK, 4:30 at 4:30. Tea would be in between 8 and 9. are halfway through 8 and 9 since it's 4:30, T equals 8.5, OK? So that means at 4:30, this is our value of T. Now how are we going to find the instantaneous growth growth rate at this time? Well, we know that the tangent line at a point represents the instantaneous growth rate. If we were to draw the tangent line here, OK, through, let's see if I can get this as properly as I can, through a point where T equals 8.5, what do you notice? We'll notice here that the tangent line is approx or the value of the slope of the tangent line would be approximately equal to the value of the slope of the C can line of the two closest points to T equals 8.5. The closest points here are when T equals 8 and T equals 9. Now, since, so we can say then that since it 2120. And 9, 2410. Are the closest points. Are the closest points to T equals 8.5. We can approximate the instantaneous population growth rate using the value of the slope of the Sean line between those points, OK? In other words, The slope M Will be the instantaneous population growth rate which will be B 9 minus B8 divided by 9 minus 8. We know those values are going to be 2410 minus 2120 divided by 1, which is equal to 290. So now this 290 is a reasonable estimate because the curve is approximately linear between T equals 8 and T equals 9. That means that the slope of the tangent line is approximately equal to. The slope of the second line, which is 290 bacteria per hour, OK? So we can say the instantaneous growth rate. Let me write that properly here. Let me spell it properly first of all, instantaneous growth rate. Is approximately. 290 bacteria. Or or. Thanks a lot for watching everyone. I hope this video helped.

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>Estimate the instantaneous rate of growth in 1985.

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The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Estimate the instantaneous rate of growth in 1985.