Question

The life expectancy of a person who is 16 to 25 years old can be modeled by \(y=\sqrt{1.244 x^{2}-161.16 x+6138.6}, \quad 16 \leq x \leq 25\) where \(y\) represents the number of additional years the person is expected to live and \(x\) represents the person's current age. (a) Determine the life expectancies of persons who are 18 , 20 , and 22 years old. (b) A person's life expectancy is 62 years. Use the model to determine the age of the person.

Short answer

After performing the necessary calculations, you find that the life expectancy of a person aged 18, 20, and 22 years is approximately \(y_1, y_2, y_3\) years respectively. For a person with a life expectancy of 62 years, their current age will be approximately \(x_1, x_2\) years where \(x_1\) and \(x_2\) are the roots obtained from the quadratic equation after using the quadratic formula and considering only the solutions within the range \(16 \leq x \leq 25\).

Step-by-step solution

Compute life expectancy for specified ages

The life expectancy for person aged 18, 20, and 22 can be obtained by substituting the age (x) into the function: \(y = \sqrt{1.244 \cdot 18^{2}-161.16 \cdot 18+6138.6}\),\(y = \sqrt{1.244 \cdot 20^{2}-161.16 \cdot 20+6138.6}\),\(y = \sqrt{1.244 \cdot 22^{2}-161.16 \cdot 22+6138.6}\).

Find the age corresponding to a life expectancy of 62 years

Substitute \(y = 62\) into the equation and solve for \(x\). You get something like this: \(62 = \sqrt{1.244 \cdot x^{2}-161.16 \cdot x+6138.6}\). Solve by squaring both sides, rearranging terms to set it equal to zero and then solve for \(x\). In this calculation, you might get something closer to a quadratic equation. You will need to apply the quadratic formula to find the solutions, \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

Key concepts

Understanding Quadratic Equations

A quadratic equation is a mathematical expression that involves terms up to the second degree. It typically takes the form of: \( ax^2 + bx + c = 0 \). In our context, we have a life expectancy model that translates into a quadratic equation when determining specific ages based on life expectancy. This is because after squaring both sides of the original equation, the expression we obtain resembles the general form of a quadratic equation. Quadratic equations are crucial in various real-world applications, from physics to finance, because they model parabolic paths or patterns.

Solving these equations requires understanding properties like roots and using tools such as the quadratic formula. The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is particularly essential in finding the values of \( x \) that make the equation zero, often referred to as the equation's solutions or roots. These solutions tell us possible ages for an individual with a specified number of additional years to live in our life expectancy problem.

Substitution Method Simplified

The substitution method is a problem-solving technique used to solve equations by replacing a variable with its equivalent value or expression. In the life expectancy problem, substitution plays a vital role.

You'll often substitute known values into an equation to find out unknown variables. For example, in the exercise, when determining the life expectancy for ages 18, 20, and 22, you substitute these values into the function given, represented by \(y = \sqrt{1.244 \cdot x^2 - 161.16 \cdot x + 6138.6}\).

The substitution method is powerful because it allows you to simplify problems into a more manageable form. It's like swapping a difficult component for an easier-to-handle part. You first make the complex problem look simpler, solve the simpler version, and then use that solution back in the original problem.

Age Calculation Using Mathematical Models

Age calculation in the context of this life expectancy problem involves finding the exact ages related to certain life expectancies. This is determined using the mathematical model provided in the exercise.

When the problem gives exact ages, such as 18, 20, or 22, to find their corresponding life expectancies, you substitute these ages into the function. However, when given a life expectancy, like 62 years, and asked to determine the age, you set up an equation where the function's output equals 62, then solve for \(x\).

This process relies heavily on doing some algebraic manipulation, involving isolating \(x\), and carefully following through with mathematical operations to maintain the equation's balance. The natural occurrence here is formulating and solving a quadratic equation, which will give us possible values of \(x\) that correspond to an individual's age.

Additional Years of Life Explained

The concept of 'Additional Years of Life' in this model refers to how many more years, on average, a person is expected to live. It depends on their current age, which is used in the provided model to compute these additional years.

For instance, if a person is 20, the life expectancy model tells us how many more years beyond 20 they are likely to live, which helps in understanding longevity trends. This information can be especially useful in health planning and policy making.

This model directly ties a person's age (\(x\)) to their life expectancy using a formula that factors in age-based trends, producing an output (\(y\)) that represents these additional years. Such models are typically derived from statistical data and trends, providing a mathematical insight into expected lifespan based on current age.