Problem 84 - Applied Calculus

Question

The usual estimate that each human-year corresponds to 7 dog-years is not very accurate for young dogs, since they quickly reach adulthood. Exercises 83 and 84 give more accurate formulas for converting human-years \(x\) into dog-years. For each conversion formula: a. Find the number of dog-years corresponding to the following amounts of human time: 8 months, 1 year and 4 months, 4 years, 10 years. b. Graph the function. The following function expresses dog-years as 15 dogyears per human-year for the first year, 9 dog-years per human-year for the second year, and then 4 dogyears per human-year for each year thereafter. \(f(x)=\left\\{\begin{array}{ll}15 x & \text { if } 0 \leq x \leq 1 \\\ 15+9(x-1) & \text { if } 1 < x \leq 2 \\ 24+4(x-2) & \text { if } x > 2\end{array}\right.\)

Step-by-Step Solution

Answer

8 months: 10 dog-years; 1 year 4 months: 18 dog-years; 4 years: 32 dog-years; 10 years: 56 dog-years.

Step 1: Convert 8 months into dog-years

First, convert 8 months into a fraction of a year: \( 8 \text{ months} = \frac{8}{12} \text{ years} = \frac{2}{3} \text{ years} \).Since \( \frac{2}{3} \leq 1 \), use the formula \( 15x \).Calculate: \( 15 \times \frac{2}{3} = 10 \text{ dog-years} \).

Step 2: Convert 1 year and 4 months into dog-years

First, convert 1 year and 4 months into years: \( 1 \text{ year} + \frac{4}{12} \text{ of a year} = 1 + \frac{1}{3} = \frac{4}{3} \text{ years} \).Since \( 1 < \frac{4}{3} \leq 2 \), use the formula \( 15 + 9(x-1) \).Calculate: \( 15 + 9 \left( \frac{4}{3} - 1 \right) = 15 + 9 \times \frac{1}{3} = 18 \text{ dog-years} \).

Step 3: Convert 4 years into dog-years

Use the formula for \( x > 2 \) which is \( 24 + 4(x-2) \).Calculate: \( 24 + 4(4-2) = 24 + 4 \times 2 = 32 \text{ dog-years} \).

Step 4: Convert 10 years into dog-years

Use the same formula for \( x > 2 \), \( 24 + 4(x-2) \).Calculate: \( 24 + 4(10-2) = 24 + 4 \times 8 = 56 \text{ dog-years} \).

Step 5: Graph the function

Graph the function by plotting the piecewise linear segments:- For \( 0 \leq x \leq 1 \), plot \( f(x) = 15x \) which is a straight line starting from the origin.- For \( 1 < x \leq 2 \), plot \( f(x) = 15 + 9(x-1) \) which starts at \( 15 \) and ends at \( 24 \).- For \( x > 2 \), plot \( f(x) = 24 + 4(x-2) \) which extends the graph as a line with a slope of 4.Use a graphing tool or manual plot to visualize this piecewise function.

Key Concepts

Dog-Years Calculation

Calculating dog-years can be interesting because it tells us how dogs age in comparison to humans. The traditional method suggests that each human year equals seven dog-years. However, this method is fairly simplistic and doesn't accurately represent how dogs, particularly young dogs, develop. Their aging process is different, especially in the early stages of life. In this exercise, a more nuanced formula is used for calculation, where the first year counts as 15 dog years due to rapid growth and development. The second year is a bit slower, adding 9 dog years. For any following years, we add 4 dog years per human year. This piecewise approach gives us a closer look at how dogs mature. The steps to calculate involve splitting human ages into sections of development and applying the corresponding dog-year conversion.

Human-to-Dog Years Conversion

The conversion of human years to dog years isn't as straightforward as one might think. Using a piecewise function simplifies the conversion by adjusting for the different stages in a dog's life.
When a dog is less than a year old, we multiply the human years by 15, as this period sees significant development.
For a dog aged between one and two years, we multiply the difference in years by 9 and add 15, acknowledging that while growth slows, they are still developing quickly. Once a dog surpasses two years of age, the growth rate decreases, converting to adding just 4 dog years per human year.
By breaking down human life stages into these distinct parts, we cater the conversion to better reflect actual dog aging.

Graphing Piecewise Functions

Graphing piecewise functions might seem complex at first, but once you break them down, they become much easier to manage. A piecewise function is defined in "pieces," or intervals, with different expressions for each segment.
For this exercise, the function has three segments:

• The first segment encompasses from 0 to 1 human years, where the function is linear with a steep slope, representing fast aging.
• The second interval, from 1 to 2 years, shows a slightly lesser slope as the aging begins to taper off.
• Finally, for all years greater than 2, the slope is more gentle, continuing the trajectory with constant growth.

To graph these segments, plot each line starting from its interval's beginning, ensuring each section's end connects smoothly to the next to represent the ongoing nature of aging.

Age Conversion Formulas

Age conversion formulas attempt to translate human years into a different perspective, such as dog years. These formulas can vary greatly based on the life stages of the species you're converting to or from. In our context, we focus on how these conversions work for dogs.
The given formula converts human years with distinct rules for each stage in a dog's life. During early stages, a multiplier of 15 is used, signifying rapid growth.
In the mid-stage, this slows to 9 as they mature. Finally, as the dog grows older, a multiplier of 4 steady the aging process, reflecting a slower pace.
These stages mimic the developmental phases a dog goes through, translating human understanding into the canine world. Complex calculations, like these in piecewise formats, allow for more realistic representations of aging across different species.