Problem 2 - Elementary Algebra

Question

It is estimated that \(t\) years from now the population of a particular city will be \(P=t^{2}-24 t+96,000\) How many years from now will the population be \(95,865 ?\)

Step-by-Step Solution

Answer

Answer: The population of the city will be 95,865 either in 9 years or in 15 years.

Step 1: Substitute the value of P in the equation

We have the equation for the population \(P\) as a function of \(t\): \(P = t^2 - 24t + 96000\) We are given that the population will be \(95,865\). So we substitute \(P\) with \(95,865\) in the equation: \(95,865 = t^2 - 24t + 96000\)

Step 2: Rearrange the equation

Now we have the quadratic equation: \(t^2 - 24t + 96000 = 95,865\) Subtract \(95,865\) from both sides of the equation to make it in the standard form: \(t^2 - 24t - (95,865 - 96,000) = 0\) Simplify the subtraction: \(t^2 - 24t + 135 = 0\)

Step 3: Solve the quadratic equation

Now, we have to solve for \(t\) in the quadratic equation: \(t^2 - 24t + 135 = 0\) To solve this equation, we can either use the quadratic formula or try to factor it. We'll use the quadratic formula here: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In our equation, \(a = 1\), \(b = -24\), and \(c = 135\). Plug the values into the formula: \(t = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 (1)(135)}}{2(1)}\)

Step 4: Simplify the quadratic formula expression

Now, we simplify the expression inside the square root and other parts of the formula: \(t = \frac{24 \pm \sqrt{576 - 540}}{2}\) \(t = \frac{24 \pm \sqrt{36}}{2}\) Since \(\sqrt{36}\) is equal to \(6\), we have: \(t = \frac{24 \pm 6}{2}\)

Step 5: Find the two possible values of t

There are two possible values of \(t\): Case 1 (addition): \(t = \frac{24 + 6}{2} = \frac{30}{2} = 15\) Case 2 (subtraction): \(t = \frac{24 - 6}{2} = \frac{18}{2} = 9\)

Step 6: Interpret the results

We have two possible values for \(t\): \(9\) years and \(15\) years. That means the population of the city will be \(95,865\) either \(9\) years from now or \(15\) years from now.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions where the highest exponent of the variable is 2. They often take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable, usually representing an unknown value we want to find.
Quadratic equations can represent a variety of physical phenomena, such as projectile motion or, as seen in this exercise, model population growth. There are several methods to solve quadratic equations:

• Factoring: Finding two binomials that, when multiplied together, give the original quadratic expression.
• Completing the square: Rewriting the equation in such a way that one side becomes a perfect square trinomial.
• Using the quadratic formula: Given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), a general solution that can find the roots directly when factoring is difficult.
• Graphing: Finding where the graph of the quadratic function intersects the x-axis.

In our step-by-step solution, the quadratic formula is used for its reliability, especially when equations seem challenging to factor.

Solving Equations

The core task in many algebra problems is solving equations, where we determine the value(s) of the variable that satisfy the equation. This process can vary significantly based on the type of equation. In the context of this exercise, we are solving a quadratic equation.
To solve the equation \( t^2 - 24t + 135 = 0 \):

• Recognize it as a quadratic equation and identify \( a = 1 \), \( b = -24 \), and \( c = 135 \).
• Apply the quadratic formula: \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Calculate the discriminant \( b^2 - 4ac \), which determines the nature of the roots (real or complex).
• Simplify the expression considering the two possible signs in the formula, providing two potential solutions.

This method finds the exact solutions, which in this exercise result in two values for the time \( t \) the population will be a specified size.

Population Growth

Understanding how to model population growth mathematically can be crucial in planning and resource allocation within a city. Population growth is often modeled using equations because these models can predict future growth based on current trends. In our specific problem, the population is modeled by a quadratic equation, \( P = t^2 - 24t + 96000 \), where \( t \) represents time in years.
Unlike exponential growth, which is common in population studies, this quadratic model includes varying rates of change over time due to the nature of a quadratic curve:

• The equation suggests the growth might initially decrease (or increase at a slower rate) since it includes a negative linear term \(-24t\).
• Eventually, the quadratic term dominates, possibly causing an increase or other modification over time.

The solution to the quadratic equation here reveals multiple points in time where the population could reach a specific number (95,865), highlighting the flexibility and multiple scenarios that quadratic models can represent.