U.S. population growth The population p(t) (in millions) of th...

Question

U.S. population growth The population p(t) (in millions) of the United States t years after the year 1900 is shown in the figure. Approximately when (in what year) was the U.S. population growing most slowly between 1925 and 2020? Estimate the growth rate in that year.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with speed. This problem says a person drives to the destination on a bicycle. The total distance covered is shown in the following image. In the image, we're given a graph that has distance traveled in kilometers on the y axis and time and hours on the X-axis. We also have a curve plotted on this graph, and it consists of four linear line segments. The first segment goes from the point of 0.25 to the point of 2.40. The next line segment goes from 2.40 to 4.50. The next line segment goes from 4.50 to the point of 6.70. And the final line segment goes from 6.70 to 7.100. And the question was to know if this journey starts at midnight, at which time interval is the speed of the bike at its minimum, and what is the speed of the bicycle? We're given 4 possible choices as our answers. For choice A, we have 5:00 to 7 o'clock and 10 kilometers per hour. For choice B, we have midnight to 2 o'clock and 7 kilometers per hour. For choice C, we have 6 o'clock to 7 o'clock and 15 kilometers per hour. And for choice D, we have 2 o'clock to 4 o'clock and 5 kilometers per hour. So we need to first determine the time interval at which the speed of the bicycle is at its minimum. And for this graph, the speed is represented by the slope of our line. So what we need to do is identify the light segment that has the smallest slope. And by looking at a graph, we see that the small slope occurs over the time in between the time interval from T equal to 2 to T equal to 4. Now, since we start at midnight, that means that um our time interval is going from 2 o'clock. 24 o'clock. Since T equal to 2, it's gonna be 2 hours after midnight, which is gonna be 2 o'clock, and then T equal to 4 it's gonna be 4 hours after midnight, which is 4 o'clock. So, this is the time interval that I am looking for. That's one half of our question answered. Now we need to determine the speed of the bicycle over this time interval. And so here, speed is going to be equal to the slope of the line, and we call that the slope of the line is just your change in your Y variable divided by your change in your X variable. So we'll denote our slope as S, which is also representing our speed, and that's going to be equal to our change in our Y values. So we'll have our final Y value of 50 kilometers. Minus our initial Y value, which is 40 kilometers. And that quantity is divided by our change in time, which is going to be 4 hours minus 2 hours. So now we just need to simplify this expression, and when we do that, this is going to be equal to 5 kilometers per hour. So that is the speed of our bicycle over this time interval. And so that's the second half of our question answered. So now that we have our answers for a question, we can look down at our answer choices, and the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

U.S. population growth The population p(t) (in millions) of the United States t years after the year 1900 is shown in the figure. Approximately when (in what year) was the U.S. population growing most slowly between 1925 and 2020? Estimate the growth rate in that year. <IMAGE>

Key Concepts

Population Growth Rate

Inflection Points

Graphical Analysis

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