Question

The life expectancy of a person who is 48 to 65 years old can be modeled by \(y=\sqrt{0.874 x^{2}-140.07 x+5752.5}, \quad 48 \leq x \leq 65\) where \(y\) represents the number of additional years the person is expected to live and \(x\) represents the person's current age. A person's life expectancy is 20 years. How old is the person?

Short answer

The person's age 'x' can be calculated by solving the quadratic equation. After solving, make sure that the value of 'x' lies within the given domain \(48 \leq x \leq 65\)

Step-by-step solution

Substitute life expectancy value into the function

Using the provided value of life expectancy \(y = 20\) years, insert it into the given function: \(20= \sqrt{0.874 x^{2}-140.07 x+5752.5}\)

Simplify the equation

The equation can then be simplified by squaring on both sides to get: \(400 = 0.874 x^2 - 140.07 x + 5752.5\)

Rearrange the equation

The equation can be rearranged to a form of quadratic equation ax^2 + bx + c = 0. Hence we get: \(0.874 x^2 - 140.07 x + (5752.5 - 400) = 0.\) So, the quadratic equation representing the problem is \(0.874 x^2 - 140.07 x + 5352.5 = 0\)

Solve the quadratic equation

The quadratic equation can be solved using the formula \(x = [-b \pm \sqrt{b^2-4ac}] / (2a)\). Where a, b and c are coefficients from the equation. Here a = 0.874, b = -140.07, c = 5352.5. Solve for \(x\) referring to the the age 'x' lies within the given domain \(48 \leq x \leq 65\)

Key concepts

Algebraic Modeling

Algebraic modeling is a powerful tool used by mathematicians, scientists, and economists to represent real-world scenarios with mathematical expressions. Essentially, it involves creating equations or functions that describe relationships between different variables. In our example, the model
\(y=\sqrt{0.874 x^{2}-140.07 x+5752.5}\), where \(y\) is the number of additional years a person is expected to live and \(x\) is their current age, is an instance of such a model. This type of algebraic modeling can be beneficial in various fields, including actuarial science and health policy planning.
When improving the exercise, it's important to illustrate the use of algebraic modeling in simplifying complex life events into manageable mathematical statements. Here, the square root and the quadratic expression inside it succinctly capture the relationship between age and life expectancy, which can be otherwise challenging to quantify. The practicality of algebraic modeling is its ability to facilitate predictions or decisions based on the outcomes of the equations.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra and is crucial for various applications in science and engineering. A quadratic equation has the standard form \(ax^2 + bx + c = 0\), with 'a' not equal to zero. The solution to these equations often requires finding the values of 'x' that satisfy the equation, which can be done using several methods such as factoring, completing the square, or utilizing the quadratic formula.
In our real-life example, we employ the quadratic formula \(x = [-b \pm \sqrt{b^2-4ac}] / (2a)\), which provides a reliable method for finding the roots of any quadratic equation. To enhance the student's understanding, it's vital to explain why we square both sides of the equation during step 2, transforming the square root form into a quadratic equation, allowing us to then apply the quadratic formula. This process embodies the mathematical strategy of manipulating an equation to make it more solvable. Furthermore, students should be encouraged to consider the domain \(48 \leq x \leq 65\), as real-life constraints often limit the acceptable range of solutions.

Mathematical Functions

Mathematical functions are expressions that uniquely associate members of one set with members of another set, often describing the relationship between input and output variables. In other words, for every input, there is exactly one output. Functions are a core concept in mathematics and are used extensively to represent a wide variety of relationships.
In the context of our exercise, the square root function encapsulates the relationship between a person's age and their life expectancy. When explaining functions, highlighting their components like the input variable (age), output variable (life expectancy), domain (range of age), and range (possible life expectancy values) can provide clarity on how functions connect different quantities. For deeper comprehension, we may discuss the inverse of a function and how, in some cases, input and output can be interchanged, underscoring the versatility of functions in mathematical modeling.