Problem 42 - College Algebra

Question

The function $$f(x)=104.5 x^{2}-1501.5 x+6016$$ models the death rate per year per \(100,000\) males, \(f(x),\) for U.S. men who average \(x\) hours of sleep each night. How many hours of sleep, to the nearest tenth of an hour, corresponds to the minimum death rate? What is this minimum death rate, to the nearest whole number?

Step-by-Step Solution

Answer

The amount of sleep corresponding to the minimum death rate is 7.2 hours and the minimum death rate is 1126 deaths per year per 100,000 males.

Step 1: Identify the coefficients

First, identify the coefficients \( a, b, c \) from the quadratic equation \( f(x) = 104.5x^2 - 1501.5x + 6016 \). Here, \( a = 104.5, b = -1501.5, c = 6016 \).

Step 2: Calculate x-coordinate of the vertex

To find the number of hours of sleep that corresponds to the minimum death rate, compute the x-coordinate of the vertex of the quadratic function using the formula \( -\frac{b}{2a} \). Substitute the values of \( b = -1501.5 \) and \( a = 104.5 \) to get \( x = -\frac{-1501.5}{2*104.5} = 7.2 \). Therefore, 7.2 hours of sleep corresponds to minimum death rate.

Step 3: Calculate y-coordinate of the vertex

To find the minimum death rate, substitute the \( x \) value obtained from step 2 into the function \( f(x) \). Hence, \( f(7.2) = 104.5*(7.2)^2 - 1501.5*7.2 + 6016 = 1126 \). Therefore, the minimum death rate is 1126 deaths per year per 100,000 males.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a special point that represents the turning point of a quadratic function. In simpler terms, it's where the function reaches either its highest or lowest value—also referred to as the maximum or minimum of the function. For a quadratic equation of the form \( ax^2 + bx + c \), the vertex can provide significant insight into the characteristics of the parabola.

The x-coordinate of the vertex can be found using the formula:

• \( -\frac{b}{2a} \)

This calculation offers us the specific x-value at which the function achieves its maximum or minimum.

In our exercise, the function \( f(x) = 104.5x^2 - 1501.5x + 6016 \) needs us to discover how many hours of sleep results in the lowest death rate. After computing, we found that the x-coordinate of the vertex is 7.2. This informs us that around 7.2 hours of sleep leads to that minimum death rate. Therefore, identifying the vertex point is crucial in understanding where the function hits its lowest point.

Quadratic Formula

The quadratic formula is a key tool for solving quadratic equations and finding the roots of the equation, which can be described as the values of \( x \) that make the equation equal zero. The standard quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It is derived from the process of completing the square and always works for any quadratic equation of the form \( ax^2 + bx + c = 0 \).While the quadratic formula itself isn't directly used in finding the vertex, it is essential for understanding why certain parabolic paths are taken and which direction they face. In our context, although it's not required to solve directly for the vertex, being familiar with it gives us a comprehensive understanding of the behavior of quadratic functions under various conditions.

The discriminant, \( b^2 - 4ac \), embedded within this formula, indicates the nature of the roots (real or complex), but that is beyond our current scope of directly finding the vertex. Keeping this formula in mind empowers you to delve deeper into a complete study of quadratic equations.

Minimum Value of a Function

Identifying the minimum value in a quadratic function is incredibly useful for understanding real-world applications such as our current problem, which deals with the death rate in relation to hours of sleep. The minimum value of the function occurs at the vertex of a parabola when the quadratic function opens upwards, as indicated by a positive \( a \) value. To find the minimum value, we compute the y-coordinate of the vertex by substituting the x-coordinate found using \( -\frac{b}{2a} \) back into the function \( f(x) \). In our exercise:

• The function \( f(x) = 104.5x^2 - 1501.5x + 6016 \)
• With an x-value of 7.2, the calculation is \( f(7.2) = 1126 \)

This tells us the minimum value, or the lowest death rate per 100,000 males, which is 1126 deaths according to the parameters given. Understanding this minimum helps to establish a crucial real-world link between sleeping habits and health outcomes, emphasizing the importance of the study of these mathematical concepts.