Question

The population, \(P\), of Town \(Y\) since 1995 can be estimated by the equation \(P=1.0635 x+3,250\), where \(x\) is the number of years since 1995 and \(0 \leq x \leq 20\). In the context of this equation, what does the number \(1.0635\) most likely represent? A) The estimated population of town \(Y\) in 1995 B) The estimated population of town \(Y\) in 2015 C) The factor by which the population of town \(Y\) increased yearly D) The factor by which the population of town \(Y\) decreased yearly

Short answer

(C) The factor by which the population of town Y increased yearly

Step-by-step solution

Understand the general form of population modeling equation

Generally, in a population modeling equation, the population P can be modeled as a function of time t: P = mt + c, where m is the population change factor and c is the initial population.

Interpret the equation for Town Y

The given equation is \( P = 1.0635x + 3250 \), where \(P\) represents the population and \(x\) represents the number of years since 1995. Here, we see that the equation takes the form of P = mt + c. So in this case, the number 1.0635 could represent the population change factor, and 3250 would represent the population of town Y in 1995.

Compare the options given A, B, C, and D

Upon comparing the given choice with the above reasoning, we find that choice (C) best matches our interpretation of the equation: "The factor by which the population of town Y increased yearly." Thus, the answer is: \( \boxed{\text{(C) The factor by which the population of town Y increased yearly}} \)

Key concepts

Mathematical Modeling

Understanding mathematical modeling is like learning a new language that describes real-world scenarios with mathematical structures. When it comes to population growth, a mathematical model aims to simulate changes in a population over time. The equation involves variables and constants: the variable often represents time, and the constants may encapsulate rates of change, initial conditions, or other factors pertinent to the population's behavior.

For instance, the equation in our example, P=1.0635x+3,250, is a simple linear model. It predicts population P based on the number of years since 1995, designated by x. To make this equation useful, it's vital to understand the significance of each component. The number 1.0635 is not arbitrary; it's the growth rate, showing how the population changes each year, while the constant term 3,250 represents the initial population in the base year.

Linear Equations

Linear equations are the simplest forms of equations in algebra and serve as the backbone for more complex mathematical concepts. They have one or more variables but no exponents higher than one, forming a straight line when graphed on a coordinate plane. A linear equation typically looks like y = mx + b, where m is the slope and b is the y-intercept.

In population modeling, linear equations can be used if the growth rate is constant. For the exercise at hand, the term 1.0635 is analogous to the slope m, indicating a steady increase in population as each year passes. The constant 3,250 is akin to the y-intercept b, suggesting that a population count started at this value at the year marked zero, which in this context is 1995.

SAT Mathematics Preparation

Effective SAT mathematics preparation involves mastering various math topics, including the principles of linear equations and mathematical modeling which are prevalent throughout the SAT Math section.

To ace questions like our population modeling example, students should focus on:

• Understanding the relationship between variables and constants in an equation.
• Recognizing the significance of the slope and y-intercept in linear equations.
• Applying logical reasoning to interpret mathematical models within the context of word problems.
• Practicing with a variety of equations to build familiarity and confidence.

Interpreting the number 1.0635 as the annual population increase rate showcases the critical thinking skills SAT questions often require. Remember, practice makes perfect, and becoming proficient at translating word problems into equations is a key step in SAT math success.