Problem 83 - Brief Applied Calculus

Question

The following function expresses dog-years as \(10 \frac{1}{2}\) dog-years per human-year for the first 2 years and then 4 dog-years per human-year for each year thereafter. \(f(x)=\left\\{\begin{array}{ll}10.5 x & \text { if } 0 \leq x \leq 2 \\\ 21+4(x-2) & \text { if } x>2\end{array}\right.\)

Step-by-Step Solution

Answer

Use 10.5x for 0-2 years; 21+4(x-2) for over 2 years.

Step 1: Understanding the Problem

We need to convert human years into dog years based on a specific rule: the first 2 human years each equal 10.5 dog years, and every year after that equals 4 dog years.

Step 2: Analyzing the Function

The function is piecewise: \( f(x) = 10.5x \) for human years \( 0 \leq x \leq 2 \), and \( f(x) = 21 + 4(x - 2) \) for human years \( x > 2 \). This reflects the change in how dog years accumulate over time.

Step 3: Calculating for 0 to 2 Human Years

If a dog is between 0 to 2 human years old, you use \( f(x) = 10.5x \). For example, at 2 human years, \( f(2) = 10.5 \times 2 = 21 \) dog years. This confirms the base case definition for the first 2 years.

Step 4: Calculating for More Than 2 Human Years

If a dog is over 2 human years old, the function is \( f(x) = 21 + 4(x - 2) \). For example, at 3 human years, \( f(3) = 21 + 4(3 - 2) = 21 + 4 \times 1 = 25 \) dog years.

Step 5: Verifying with Different Values

Try a few more values. For example, if a dog is 5 human years old, \( f(5) = 21 + 4(5 - 2) = 21 + 12 = 33 \) dog years. This shows the pattern of adding 4 dog years for every additional human year beyond the first two.

Key Concepts

Dog Years Calculation

When we talk about dog years, we're essentially discussing how to translate the age of a dog into terms that better reflect their life stages, compared to human years. This conversion helps in understanding a dog's development and aging process in a more relatable way for humans. The given function in this exercise allows us to calculate the dog's age in dog years, based on the number of human years. To calculate dog years:

• The first 2 human years are considered to be quite significant in a dog's life, where each year counts as 10.5 dog years.
• After the initial two years, each subsequent human year is equivalent to 4 dog years.

Breaking this into components makes it easier to see how a dog's age changes dramatically in the first years and stabilizes after that.

Human Years to Dog Years Conversion

Converting from human years to dog years using the given piecewise function ensures a more accurate representation of a dog's life stages. This conversion can be understood through two distinct phases:**Phase 1: Initial Two Years**

• For the first two human years, the conversion formula is simple: Multiply the human years by 10.5.
• For instance, if a dog is 1 human year old, then using the function, we get: \( f(1) = 10.5 \times 1 = 10.5 \) dog years.

**Phase 2: Years Beyond Two**

• Starting from the third year, the conversion changes: It involves a base of 21 dog years for the first two human years, plus an additional 4 dog years for each extra human year.
• Thus, at 3 human years, a dog is \( f(3) = 21 + 4(3 - 2) = 25 \) dog years.

These phases reflect the rapid growth and aging of dogs in their early years, followed by a slower aging process.

Mathematical Modeling

Mathematical modeling uses mathematical expressions to represent real-life situations. In this context, our problem employs a piecewise function to model how dogs age differently compared to humans. **Understanding Piecewise Functions**

• A piecewise function is a mathematical tool that channels different conditions or rules into a single function.
• In our dog years calculation, the function is divided into two rules, reflecting the faster aging in early life vs. slower aging in later years.

The power of this modeling approach lies in its precision, capturing the nuances in how dogs age. By employing separate calculations for different life stages, it offers an accurate and simple means to estimate the age of dogs. This practical application of mathematics showcases how versatile and impactful mathematical modeling can be.