Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

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 (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.

Step-by-Step Solution

Answer

(a) The greatest population according to the model was achieved in the year derived in step 1 and the population figure obtained in step 2. (b) The population in 2075 is estimated to be the result found in step 3. The reliability of this figure can be questionable, as noted in step 4, since it extrapolates far beyond the time range for which the model was initially designed.

Step 1: Finding the Year With the Greatest Population (a)

The year with the greatest population corresponds to the maximum point of the quadratic function \(P(t)\), within the range 0<=t<=13. For a quadratic function in the form \(ax^2 + bx + c\), the maximum point occurs at \(x = -b/2a\). Inserting the coefficients into the formula gives, \(t = - (95.9)/(2*(-14.82))\)

Step 2: Computing for the Greatest Population (a)

After obtaining the year with the greatest population, substitute this value into the \(P(t)\) function to find the corresponding population. Hence, \(P(t) = -14.82*t^{2} + 95.9*t + 82276\)

Step 3: Determining the Population in 2075 (b)

To determine the population in 2075, substitute \(t = 2075 - 2000\) into the function \(P(t)\)

Step 4: Analyzing Reasonableness of the Result (b)

For this step, comment on the reasonableness of the model's prediction. Note that the function is only certain within the range 0<=t<=13. Hence, predicting the population for \(t = 2075 - 2000\) extends far beyond its reliable range which might result in an inaccurate prediction.

Key Concepts

Population Modeling

Population modeling uses mathematical functions to represent how populations change over time. In this exercise, we observe Germany's population modeled by a quadratic function, which looks like a parabola when graphed. Quadratic functions often effectively simulate populations over specific time frames due to their ability to capture initial growth and eventual decline. They are defined by:

• Initial population size, indicated by the constant term (here, 82,276 thousands).
• The rate and curvature of change, shown by the coefficients of the linear and quadratic terms.

Models help predict future trends and understand past behaviors, offering insights but not certainties. Real-world factors such as immigration, birth rates, and policies can influence actual populations distinctly from the model's predictions.

Vertex Formula

The vertex formula is a key tool in finding the maximum or minimum point of a quadratic function, often representing either a peak or a trough in population models. The formula is:\[ x = -\frac{b}{2a} \]For our model of Germany's population:

• The quadratic coefficient \(a\) is \(-14.82\) and the linear coefficient \(b\) is \(95.9\).
• Using the vertex formula: \( t = -\frac{95.9}{2 \times (-14.82)} \).

This calculation helps determine when the population reached its highest point between 2000 and 2013, aligning with the model’s peak. By finding the vertex, we gather vital insights into implicitly capturing trends not immediately obvious from data points alone.

Extrapolation

Extrapolation involves using a model to predict data points outside the established range of known values. While this can provide forecasts, such as projecting Germany's population in 2075 using:\[ t = 2075 - 2000 = 75 \]It's important to exercise caution; models are built based on existing data and assumptions that could change significantly in the long term. As this model is only validated from 2000 to 2013, predicting 62 years beyond this range may yield unreliable results. Always consider the assumptions and limits of your data to gauge the feasibility of such predictions, especially with complex societal factors at play.