Average single-family home prices P (in thousands of dollars) ...

Question

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.

Graph showing average home prices in Sacramento from 2006 to 2015, with a notable decline around 2008.

d. During what year did home prices drop most rapidly and what is an estimate of this rate?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The population of a small town from 1989 to 1999 is shown using the given graph shown below. During which year did the population drop the most rapidly and what was the rate of this decrease? Awesome. So it appears for this particular prom. We're asked to solve for two separate answers. Our first answer we're ultimately trying to solve is we're trying to determine during which year did the population drop the most rapidly. That's our first answer, and our second answer is we're trying to figure out what was the rate of this decrease. That's our second and final answer we're ultimately trying to solve for. So with that said, let's quickly look at our graph to see what exactly is happening. So as you can see, we have our standard coordinate plane where our Y axis represents the population P in units of thousands, and our X-axis represents T, the time in units of years. And then we have our blue curve representing the change in population over the years from 1989 to 1999. And as you can see, our population starts high, then it starts to decrease, decrease, decrease until it levels out a little bit, and then it slowly starts to increase, increase, increase. But it levels out for a little bit, it appears. And once again, our curve represents the change in population throughout the years. So now that we have a good visualization of what's happening for this particular prom, let's read off our multiple choice answers to see what our final answer pair might be. A is 1991, 80,000 people per year, B is 1992, 70,000 people per year. C is 1991, 60,000 people per year, and finally, D is 1991, 65,000 people per year. Awesome. OK. So first off, let us note that the need to note that the most rapid decline occurred during the year 1991 because the slope of the curve is steepest that year, indicating the fastest decrease in population. So let's circle that. So this is 1991, so let's make a note of that. So, 1991. Awesome. So once again, the most rapid decline occurred during the year 1991 because the slope of our curve is the steepest for that year, indicating the fastest decrease in population. Also, from looking at our graph, we can see that at the start of 1991, the population was around 260,000, since as you could see on the graph. So it's around 260,000, and at the end of 1991, so at 1991, at the end, the population fell to 180,000. So, that means that the change of population will be 180,000. Minus 260,000, which is going to be equal to negative 80,000. Since it's decreasing rapidly, so it's gonna be a negative number because the population is decreasing. That's what that negative symbol means. So, let us also note that the time interval is also equal to, so it's called this time TI time interval is also equal to 1 year. Fantastic. So that means that the rate of decrease, so we'll call it R. the rate of decrease, so the rate of decrease is going to be equal to negative 80,000 divided by 1. So that means it's going to be simply equal to negative 80,000, and not just any, it's it's not unitless, it's technically negative 80,000. People Per year, since that's our unit, because as you could see. We have 80,000 people. And then our units down below, since we know that the time interval is one year, so it'll be people per year, so people per year. And that's it. We've officially solved for this problem. So that means that our final answer or answers, I should say, without a doubt, have to be multiple choice answers or multiple choice answer letter A because it has our final two answers for. The year, which is 1991, and our rate of this decrease is 80,000 people per year. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.

d. During what year did home prices drop most rapidly and what is an estimate of this rate?

Key Concepts

Rate of Change

Local Maximum and Minimum

Secant Line

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