Problem 84 - Brief Applied Calculus

Question

GENERAL: Longevity When a person reaches age 65 , the probability of living for another \(x\) decades is approximated by the function \(f(x)=-0.077 x^{2}-0.057 x+1 \quad\) (for \(\left.0 \leq x \leq 3\right)\) Find the probability that such a person will live for another: a. One decade. b. Two decades. c. Three decades.

Step-by-Step Solution

Answer

a. 86.6% b. 57.8% c. 13.6%

Step 1: Understanding the Function

The function given is \( f(x) = -0.077x^2 - 0.057x + 1 \). This function approximates the probability of a person, who is currently 65 years old, living for another \(x\) decades, where \(0 \leq x \leq 3\). Our task is to find \(f(x)\) for different values of \(x\): 1, 2, and 3.

Step 1: Calculate for One Decade

To find the probability that the person will live another one decade, evaluate the function at \(x = 1\). Substitute \(x = 1\) into the function: \[ f(1) = -0.077(1)^2 - 0.057(1) + 1 \] Calculate: \[ f(1) = -0.077 - 0.057 + 1 = 0.866 \] Thus, the probability of living another decade is 0.866 or 86.6%.

Step 2: Calculate for Two Decades

To find the probability that the person will live another two decades, evaluate the function at \(x = 2\). Substitute \(x = 2\) into the function: \[ f(2) = -0.077(2)^2 - 0.057(2) + 1 \] Calculate: \[ f(2) = -0.308 - 0.114 + 1 = 0.578 \] Thus, the probability of living another two decades is 0.578 or 57.8%.

Step 3: Calculate for Three Decades

To find the probability that the person will live another three decades, evaluate the function at \(x = 3\). Substitute \(x = 3\) into the function: \[ f(3) = -0.077(3)^2 - 0.057(3) + 1 \] Calculate: \[ f(3) = -0.693 - 0.171 + 1 = 0.136 \] Thus, the probability of living another three decades is 0.136 or 13.6%.

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions that include a term with a variable squared, typically written as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. These types of functions are important because they can model a wide range of real-world phenomena, including projectile motion and certain types of growth and decay.

In the context of the given probability function \( f(x) = -0.077x^2 - 0.057x + 1 \), the expression is a quadratic function with respect to the variable \( x \), which represents the number of decades. Here, the coefficients \(-0.077\) and \(-0.057\), along with the constant term \(1\), determine the shape and position of the quadratic graph. Quadratic functions generally create a parabolic shape on a graph. Because the coefficient \(-0.077\) of the \(x^2\) term is negative, the parabola opens downwards. This feature indicates that the function will peak at a certain point and then decrease as \( x \) increases. This general behavior is useful in understanding how the probability changes over time.

Function Evaluation

Function evaluation involves determining the output of a function given a specific input. It's a foundational mathematical concept that allows us to discover the value of a function for particular values of its variable.

Consider the example from the exercise, where we evaluate \( f(x) = -0.077x^2 - 0.057x + 1 \) for different values of \( x \) to find the respective probabilities. For \( x = 1 \), we substitute 1 into the function: \( f(1) = -0.077(1)^2 - 0.057(1) + 1 = 0.866 \). This calculation tells us that the probability of a 65-year-old living one more decade is 86.6%. A similar process is done for \( x = 2 \) and \( x = 3 \), reflecting the decreasing probabilities over extended periods.

This exercise in function evaluation is crucial as it demonstrates how mathematical models can provide estimates for real-world scenarios. It highlights the tangible connection between input variables and output results, turning abstract functions into meaningful predictions.

Life Expectancy Calculations

Life expectancy calculations are a significant application of probability and mathematical functions. These models help estimate how long people are likely to live based on certain age transitions and demographic data. In this exercise, we're using a given quadratic function to predict the additional life expectancy for someone who is already 65 years old.

The function \( f(x) = -0.077x^2 - 0.057x + 1 \) specifically predicts the probability of living additional decades beyond age 65. Such calculations are valuable in various fields, including healthcare planning, insurance, and social services, as they provide insights into population longevity and potential needs over time.

By evaluating \( f(x) \) for different decade markers, we can interpret that as the number of additional decades increases, the probability of surviving through those decades decreases. This reflects real-world aging trends, where the likelihood of living through longer future spans diminishes as one grows older. Understanding this type of probability calculation can be essential for students and professionals dealing in statistic-heavy fields or developing policies based on demographic projections.