Question

Populations in rural communities have declined in Western Canada, while populations in larger urban centers have increased. This is partly due to expanding agricultural operations and fewer traditional family farms. demographer uses a polynomial function to predict the population, \(P,\) of a town \(t\) years from now. The function is \(P(t)=t^{4}-20 t^{3}-20 t^{2}+1500 t+15000\) Assume this model can be used for the next 20 years.a) What are the key features of the graph of this function? b) What is the current population of this town? c) What will the population of the town be 10 years from now? d) When will the population of the town be approximately 24 000?

Short answer

Current population is 15,000. Population in 10 years is 26,500. The population will be approximately 24,000 in about 8 years.

Step-by-step solution

- Identifying Key Features of the Function

To identify key features of the graph such as intercepts, turning points, and end behaviors, analyze the polynomial function by finding its derivative and solving for critical points. Also, consider basic graph characteristics like degree and leading coefficient.

- Find the Y-intercept (Current Population)

To find the current population, substitute most recent year's value, t = 0, into the polynomial. The current population is given by P(0).

- Calculate Population 10 Years from Now

Substitute t = 10 into the polynomial to find the town's population 10 years from now. Compute P(10).

- Determine when the Population will be approximately 24,000

Set P(t) equal to 24000 and solve for t. This is a numerical method problem since it might require iterative methods or graphing calculators to approximate the value of t.

Key concepts

polynomial function

Let's start by understanding the polynomial function used to predict the population. A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this exercise, we have:
P(t) = t^4 - 20t^3 - 20t^2 + 1500t + 15000
This function is of degree 4, the highest power of t. Degree 4 polynomials are quartic functions and typically exhibit a 'W' shape or an inverted 'W' shape in their graphs. The leading coefficient here is 1 (coefficient of t^4), indicating the graph rises to positive infinity at both ends of the x-axis.

It's important to note that polynomial functions can model real-world phenomena, such as population changes, where changes are not straightforward and can have multiple turning points.

derivative

The derivative of the polynomial function helps us understand the rate of change of the population over time and locate critical points. The derivative of P(t) is found by differentiating each term concerning t:
P'(t) = 4t^3 - 60t^2 - 40t + 1500
This resulting derivative function gives us the slope of the tangent line to the curve of P(t) at any point t. By setting P'(t) = 0, we can find points where the slope is zero, which correspond to the local maxima, minima, or inflection points in the graph of P(t). These are known as critical points.

critical points

Critical points are the values of t where the derivative of the function equals zero or is undefined. These points help identify where the population changes direction in its growth or decline.
To find the critical points of P(t), we set the derivative P'(t) = 0 and solve for t:
4t^3 - 60t^2 - 40t + 1500 = 0
Solving this cubic equation can be complex, often requiring numerical methods or software tools. Once the critical points are found, we can analyze the second derivative to determine the concavity of the function at these points and classify them as local maxima, minima, or points of inflection.
Finding these points enables planners to predict periods of population growth or decline more accurately.

intercepts

Intercepts are crucial to understanding where the polynomial function crosses the axes. For population predictions, the most relevant intercept is the y-intercept, which gives the population at the time t = 0.
To find the y-intercept, substitute t = 0 into the population function P(t):
P(0) = 0^4 - 20(0)^3 - 20(0)^2 + 1500(0) + 15000 = 15000.
This means the current population of the town is 15,000.
The x-intercepts are the points where the population of the town hits zero, ideally never in this context. To find x-intercepts, solve the equation P(t) = 0.
The population prediction model assumes it never hits zero within our timeline but understanding intercepts helps outline the full behavior of the function.

population prediction

Using the polynomial function helps predict population changes over time. For example, to find the population in 10 years, substitute t = 10 into the function:
P(10) = 10^4 - 20(10)^3 - 20(10)^2 + 1500(10) + 15000
Simplifying this gives the estimated future population.
Predictions are practical, especially when identifying significant future values, such as population meeting a specific target like 24,000. To find such points, set P(t) = 24000 and solve for t:
This can be done using numerical methods—trial and error or graphing.
It involves finding when the polynomial equals the target population.
Polynomial models allow demographers to make essential forecasts and prepare for future growth or decline, shaping policies and resources allocation accordingly.