Problem 102 - Precalculus Essentials

Question

The bar graph shows your chances of surviving to various ages once you reach \(60 .\) (GRAPH CANNOT COPY) The functions \(\begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned}\) model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve. a. Find and interpret \(f(70)\) b. Find and interpret \(g(70)\) c. Which function serves as a better model for the chance of surviving to age \(70 ?\)

Step-by-Step Solution

Answer

According to the models, a 60-year old person has a 78% chance of surviving to age 70 based on function \(f\), and a 75% chance based on function \(g\). Thus, function \(f(x)\) serves as a better model for the survival chance at age 70.

Step 1: Calculate \(f(70)\)

Substitute \(x = 70\) in the \(f(x)\) equation: \(f(70) = -2.9*70 + 286 = 78\)

Step 2: Interpret \(f(70)\)

The value of \(f(70)\) represents the chance, as a percentage, that a 60-year-old will survive to age 70 according to the model \(f(x)\). So, the model suggests a 78% chance of survival.

Step 3: Calculate \(g(70)\)

Similarly, substitute \(x = 70\) in the \(g(x)\) equation: \(g(70) = 0.01*(70)^2 - 4.9*70 + 370 = 75\)

Step 4: Interpret \(g(70)\)

The value of \(g(70)\) represents the chance, as a percent, that a 60-year-old will survive to age 70 according to the model \(g(x)\). So, the model suggests a 75% chance of survival.

Step 5: Compare the Models

If we compare the results of the functions \(f(70)\) and \(g(70)\), it is clear that the function \(f(x)\) predicts a higher survival chance - 78% compared to 75% from \(g(x)\). Thus, if we interpret higher chance as a better model, function \(f(x)\) is a better model at age 70.

Key Concepts

Understanding Linear Functions

Mathematical modeling often uses linear functions due to their simplicity and directness. A linear function represents a relationship in which variables change at constant rates relative to one another. The equation of a linear function is generally expressed in the form \(f(x) = mx + b\). Here, \(m\) stands for the slope, indicating the rate of change, while \(b\) is the y-intercept, where the line crosses the y-axis.

In the given problem, the function \(f(x) = -2.9x + 286\) models the survival chance of a 60-year-old to future ages. The slope \(-2.9\) suggests a decrease in survival probability by 2.9% for each year increase in age, showing a linear decline. This type of function is beneficial when relationships remain consistent over time.

Key points about linear functions:

• Simplistic representation of change.
• Consistent rate of variation indicated by the slope.
• Commonly used in scenarios with uniform changes over time.

Exploring Quadratic Functions

Quadratic functions provide a way to model scenarios where change does not occur at a constant rate. This function type is represented by the formula \(g(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The term \(ax^2\) introduces curvature, enabling the model to capture accelerating or decelerating changes.

For this problem, \(g(x) = 0.01x^2 - 4.9x + 370\) hypothesizes the survival percentage. The presence of \(x^2\) indicates a non-linear trend, suggesting that survival probabilities fluctuate as age increases, possibly slowing the decrease rate as individuals age. Quadratic functions are relevant when changes aren't constant and may change in rate or reflect fluctuations overall.

Some key characteristics of quadratic functions include:

• Non-linear pattern representation through \(x^2\).
• Captures complex patterns of change over time.
• Illustrates scenarios with increasing or decreasing acceleration effects.

Aging Population Predictions

Predicting survival rates is a vital task in demographics and public planning, with models like linear and quadratic functions offering different predictive insights. Understanding these can help interpret population trends among elder age groups and make informed decisions in health care and resources.

Linear models like \(f(x)\) give a straightforward outlook by assuming constant change, which can be simple but sometimes oversimplifies real-world dynamics. In contrast, quadratic models such as \(g(x)\) allow for a nuanced view by considering variability in survival chances due to various health, social, and environmental factors.

Aging population predictions are crucial for:

• Planning healthcare services and facilities.
• Developing social security policies.
• Forecasting economic needs of an aging populace.

Analyzing which model offers a more accurate reflection of real-world trends depends on the specific characteristics and observed data of the population. Thus, analyzing and comparing different models helps in understanding better which to apply for predicting aging population trends.