Problem 84 - College Algebra

Question

The polynomial \(104.5 x^{2}-1501.5 x+6016\) describes the death rate per year, per \(100,000\) men, for men averaging \(x\) hours of sleep each night. Evaluate the polynomial for \(x=10 .\) Describe what the answer means in practical terms.

Step-by-Step Solution

Answer

The death rate per year per 100,000 men who average 10 hours of sleep each night is 451.

Step 1: Substitute \(x\) with \(10\)

First, replace every occurrence of \(x\) in the polynomial \(104.5 x^{2}-1501.5 x+6016\) with \(10\). You would be left with \(104.5 * 10^2 - 1501.5 * 10 + 6016\).

Step 2: Evaluate the Expression

We now carry out the calculations. So, \(104.5 * 10^{2} = 10450\), \(-1501.5 * 10 = -15015\), leaving us with \(10450 - 15015 + 6016\).

Step 3: Compute the Final Result

Adding up \(10450 - 15015 + 6016\) gives us \(451\). Thus, when \(x = 10\), the polynomial evaluates to \(451\). In practical terms, this means that if a man sleeps an average of \(10\) hours each night, the death rate per year per \(100,000\) men is estimated to be \(451\).

Key Concepts

Death Rate Estimation

Death rate estimation involves calculating the number of deaths that occur within a specific population and timeframe. In this exercise, the death rate is calculated per year for every 100,000 individuals, specifically focusing on the impact of sleep habits on men's health. Understanding these estimations can help identify health risks or benefits associated with different sleep patterns. To break it down:

• The death rate provides insights into public health and medical research.
• Estimations can guide healthcare policies and preventive measures.
• Knowing the death rate helps compare various factors like age, lifestyle, or geographical locations.

In practical terms for this problem, a death rate of 451 means that among 100,000 men who sleep an average of 10 hours per night, around 451 are predicted to pass away within a year.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables, each multiplied by a coefficient. They are represented in a standard form like this: \[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where each term consists of a constant multiplied by a power of the variable \(x\). In our case, the polynomial is \(104.5x^2 - 1501.5x + 6016\).Key points about polynomial functions include:

• They can represent various real-world phenomena, such as physical, economic, and biological systems.
• The degree of a polynomial is determined by the highest power of the variable. Here it is 2, making it a quadratic polynomial.
• Higher powers of \(x\) significantly influence the curve's shape formed by the polynomial's graph.

Polynomial functions can efficiently model trends and predict future values based on current or past data.

Substitution Method

The substitution method is a straightforward approach when evaluating functions at specific values of the variable involved. By substituting a particular value for the variable, we can calculate the function's output, which is especially useful for polynomials.Here's a simple breakdown:

• Locate the variable in the polynomial, in this case, it's \(x\).
• Replace \(x\) with the given numerical value, which is 10 in this problem.
• Perform the operations following mathematical rules (order of operations), solving until the expression is fully simplified.

In this exercise, substituting \(x = 10\) into the polynomial allows us to compute the relevant death rate, resulting in the value 451. The substitution method makes it easy to evaluate polynomial functions at specific points to derive practical insights.