The population P of Germany (in thousands) from 2000 through...

The population $P$ of Germany (in thousands) from 2000 through 2013 can be modeled by $$P(t)=-14.82 t^{2}+95.9 t+82,276, \quad 0 \leq t \leq 13$$ where $t$ is the year, with $t=0$ corresponding to $2000 .$ (Source: U.S. Census Bureau) a. According to the model, in what year did Germany have its greatest population? What was the population? b. According to the model, what will Germany's population be in the year $2075 ?$ Is this result reasonable? Explain.

The population $P$ of Germany (in thousands) from 2000 through 2013 can be modeled by $$P(t)=-14.82 t^{2}+95.9 t+82,276, \quad 0 \leq t \leq 13$$ where $t$ is the year, with $t=0$ corresponding to $2000 .$ (Source: U.S. Census Bureau)
a. According to the model, in what year did Germany have its greatest population? What was the population?
b. According to the model, what will Germany's population be in the year $2075 ?$ Is this result reasonable? Explain.

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial typically written in the form ax² + bx + c. Its graph is a parabola, which opens downward if the coefficient a is negative and upward if a is positive. In modeling scenarios, quadratic functions can capture trends that increase and then decrease or vice versa over a given interval.

Vertex of a Parabola

The vertex of a parabola is the point at which the function reaches its maximum or minimum value. For a quadratic function in the form ax² + bx + c, the vertex occurs at x = -b/(2a). This point is crucial in contexts like population modeling for identifying when a peak or a trough is reached in the data.

Population Modeling

Population modeling involves using mathematical functions to represent the growth and decline of a population over time. Models like quadratic functions are often used to fit historical data and can help identify trends such as maximum population levels. However, these models are usually valid only within a specific interval of time, and care must be taken when applying them outside that range.

Extrapolation

Extrapolation is the process of using a mathematical model based on existing data to predict values outside the range of the observed data. While it can provide short-term forecasts, extrapolation is often unreliable for long-term predictions because it assumes that the underlying trends remain constant, ignoring real-world complexities and changes that might occur over time.

Transcript

0:00 Hi everybody.

00:01 So today we're going to be doing a quick algebra problem that involves two parts.

00:06 So here we're given a model of a population of p is a population in terms of t, which is the number of years starting from the year 2000.

00:18 So the first part is we want to figure out according to this model what year had the greatest population.

00:24 So it's a parabola that opens downward.

00:28 So we're going to do this.

00:30 By finding the vertex of this problem.

00:33 So to start off, we're going to figure out when t or when p of t equals zero, what year t is.

00:44 So we'll just do zero equals.

00:48 We'll just copy and paste this.

00:51 You'll notice this is not a very easy one to solve, so we're going to use the quadratic formula.

00:56 So negative b plus square root of b squared minus 4 times a times c and all of that over 2 times a so this gives us the first year which is negative 71 .35 or 3 4 years ago from the year 2000 now the other half is that it's minus the square root so we just switch that and we get 77 .8 as our next year that the population reaches zero.

01:59 So now we're just going to add these two numbers together and divide by two.

02:11 We'll get the halfway point of 3 .24.

02:17 Now comes the fun part.

02:18 We get to plug it back into this equation here.

02:33 So i'll just quickly plug it all in...

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