Problem 12 - Elementary Algebra

Question

It is estimated that \(t\) years from now the population of a certain city will be \(P=t^{2}-15 t+12,036\) (a) What is the population now? (b) How many years from now will the population be 12,000 ?

Step-by-Step Solution

Answer

\(P(t) = t^{2} - 15t + 12,036\) Answer: The current population is 12,036, and the population will be 12,000 in 3 years and 12 years from now.

Step 1: Evaluate the population function at t=0

To find the population now, we need to substitute t=0 in the given function: \(P(0) = (0)^{2} - 15(0) + 12,036\) Now, simplify: \(P(0) = 0 - 0 + 12,036\) \(P(0) = 12,036\)

Step 2: Answer (a)

The population now is 12,036.

Step 2: Solve the equation for t when P = 12,000

To find the number of years until the population becomes 12,000, we need to solve the equation: \(12000 = t^{2} - 15t + 12,036\) Rearrange the equation: \(t^{2} - 15t + 12,036 - 12,000 = 0\) Simplify the equation: \(t^{2} - 15t + 36 = 0\) Now, factor the equation: \((t - 3)(t - 12) = 0\)

Step 3: Find the solutions for t

To find the values of t, we need to solve the equation for both factors: \(t - 3 = 0 \Rightarrow t = 3\) \(t - 12 = 0 \Rightarrow t = 12\) There are two possible solutions: t=3 and t=12 years from now.

Step 5: Answer (b)

The population will be 12,000 in 3 years and 12 years from now.

Key Concepts

Algebraic Expressions

In the realm of algebra, understanding algebraic expressions is foundational. These are made up of numbers, variables (like the 't' in our example), and arithmetic operations such as addition, subtraction, multiplication, and powers.

An expression like \(P=t^{2}-15t+12,036\) is algebraic. It represents a relation where the population 'P' changes with time 't'. Evaluating an expression is straightforward when specific values are substituted for the variables, as seen when we calculated the current population by setting \(t=0\).

Significance in the Real World

Algebraic expressions aren't abstract concepts locked in textbooks—they're tools for modeling real-life scenarios, like predicting population changes. By mastering algebraic expressions, students learn to distill complex situations into manageable, mathematical terms.

Factoring Polynomials

Polynomials are expressions consisting of variables and coefficients, brought together by addition, subtraction, multiplication, and non-negative integer exponents of variables. Factoring polynomials is a critical skill, enabling one to break down a complex expression into simpler components—and it's like finding the original pieces that, when multiplied together, give the starting polynomial.

In our population example, we transformed the quadratic polynomial \(t^{2} - 15t + 36\) into its factored form \((t - 3)(t - 12)\).

Why Factoring Matters

Factoring renders otherwise intractable algebraic equations solvable. It paves the way for easier arithmetic operations and provides deeper insights into the properties and solutions of equations—like revealing the years when population hits a certain number.

Solving Quadratic Equations

A quadratic equation is a polynomial equation in which the highest exponent of the variable is 2. The general form is \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are coefficients, and typically, 'x' represents the variable.

Solving these equations might involve factoring, completing the square, or using the quadratic formula. Factoring was our method of choice in our example, leading to two possible values for 't'. Each corresponds to a point in time when the population of the city hits 12,000.

Practical Applications

Mastering the solution of quadratic equations equips students with problem-solving skills applicable in various fields including physics, engineering, and economics. It's not just about finding 'x'—it's about understanding how different factors in a scenario interact to influence an outcome.