Problem 13 - Elementary Algebra

Question

It is estimated that \(t\) years from now the population of a certain city will be \(P=t^{2}-16 t+24,060\) (a) What is the population now? (b) How many years from now will the population be \(24,000 ?\)

Step-by-Step Solution

Answer

Answer: The current population is 24,060, and it will take 10 years for the population to reach 24,000.

Step 1: Part (a) - Finding the current population

To find the current population, set \(t = 0\) in the equation of \(P\): \(P = t^2 - 16t + 24,060\) Since \(t=0\), \(P = 0^2 - 16(0) + 24,060\) Now, simplify and solve for \(P\).

Step 2: Solving for current population

By plugging in \(t = 0\) into the equation, we simply get the value of \(P\): \(P = 0 - 0 + 24,060\) \(P = 24,060\) So, the current population is \(24,060\).

Step 3: Part (b) - Finding the number of years to reach population 24,000

To find the number of years to reach the population of \(24,000\), set \(P = 24,000\) in the equation of \(P\): \(24,000 = t^2 - 16t + 24,060\) Now, solve for \(t\).

Step 4: Solving for t when P = 24,000

Rearrange the equation to form a quadratic equation: \(t^2 - 16t + 24,060 - 24,000 = 0\) And reduce the equation to: \(t^2 - 16t + 60 = 0\) To solve the quadratic equation, we can use the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Here, \(a = 1, b = -16,\) and \(c = 60\). Now, plug in the values and solve for t.

Step 5: Applying the quadratic formula

First, find the discriminant: \(D = b^2 - 4ac\): \(D = (-16)^2 - 4(1)(60)\) \(D = 256 - 240\) \(D = 16\) Now, use the quadratic formula: \(t = \frac{-(-16) \pm \sqrt{16}}{2(1)}\) So, \(t = \frac{16 \pm 4}{2}\) Two solutions can be found: \(t_1 = \frac{16+4}{2} = \frac{20}{2} = 10\) \(t_2 = \frac{16-4}{2} = \frac{12}{2} = 6\) Since time cannot flow backwards, we will accept the larger value of \(t\), which is 10 years. So, the population will reach \(24,000\) in 10 years.

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is crucial for various mathematical analyses, including predicting future events based on models. Typically represented in the standard form of \( ax^{2} + bx + c = 0 \), these equations have a squared term as their highest degree, which results in a parabolic graph when plotted. One of the most common methods to solve them is by using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \

This formula offers a systematic approach by plugging in the values for \(a\), \(b\), and \(c\), which are coefficients from the quadratic equation. Using this formula, it is possible to determine the roots or solutions of the equation, which may be real or complex numbers. The determinant of the type of solutions we'll receive is found by evaluating the discriminant \( b^2 - 4ac \). If the discriminant is positive, there will be two distinct real solutions; if zero, one real solution; and if negative, complex solutions.

The Discriminant's Role

It is essential to compute the discriminant before applying the quadratic formula. As seen in Step 5 of the solution, \(D = b^2 - 4ac\) helps determine the nature of the equation's roots. In the population prediction exercise, the discriminant was 16, indicating that the equation has two real and different solutions, signifying two points in time where the population could be 24,000.

Population Prediction

Population prediction exercises, like the one in our example, use algebraic models for forecasting future demographics. These predictions are based on trends and assumptions that translate into an algebraic form, often using quadratic equations. In this case, predicting the population involves finding specific values for \(t\), the time variable, within the quadratic equation \(P = t^{2} - 16t + 24,060\).

These predictions are important in urban planning, resource management, and policy making, helping planners prepare for future demands. A part (a) of the exercise assesses the existing condition, meaning the current population. Part (b) requires us to predict when the population will be a specific future value, a common task for those in the field of demographic studies.

Interpreting the Results

After solving using the quadratic formula, two possible points in time were obtained. The interpretation of these results requires understanding that while the mathematics accurately models the situation, the larger solution (10 years) is the most valid prediction since time cannot be negative or reverse, as indicated by the smaller solution (6 years).

Algebraic Expressions

Algebraic expressions are the building blocks for forming equations and inequalities used to solve various problems across mathematics and its applied fields. An expression helps to abstract and represent relationships using numbers, variables, and arithmetic operations without an equality sign. In the given population prediction model, \(P = t^{2} - 16t + 24,060\) is the algebraic expression that represents the population \(P\) in terms of the time \(t\).

Understanding algebraic expressions is essential in converting word problems and real-world phenomena into solvable mathematical formats. These expressions are manipulated using arithmetic rules and operations, allowing us to resolve complex problems methodically.

The Significance of Each Component

In our example, the coefficients and constant term (16 and 24,060) help define the population model's growth curve characteristics. Insightfully, the constant term (24,060) directly indicates the city's current population when time \(t = 0\), demonstrating how every part of an algebraic expression can hold a significance beyond mere numbers.