Problem 83 - Applied Calculus

Question

83-84.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 \(10 \frac{1}{2}\) 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

Dog-years: 8 months = 7, 1 year 4 months = 14, 4 years = 29, 10 years = 53.

Step 1: Convert Human Time to Years

Convert each given human time into years. Since 1 year equals 12 months, for 8 months, the human years will be \( \frac{8}{12} = \frac{2}{3} \) years. For 1 year and 4 months, the human years will be \( 1 + \frac{4}{12} = 1 + \frac{1}{3} = \frac{4}{3} \) years.

Step 2: Apply the Conversion Formula

For converting human-years into dog-years, use the piecewise function defined in the exercise:\[f(x)=\begin{cases} 10.5x & \text{if } 0 \leq x \leq 2 \ 21+4(x-2) & \text{if } x > 2 \end{cases} \]For 8 months \( \left(\frac{2}{3} \text{ human years}\right)\) and 1 year 4 months \( \left(\frac{4}{3} \text{ human years}\right)\), use the first case since both are less than 2 years. For 4 years and 10 years, use the second case as both are greater than 2 years.

Step 3: Calculate Dog-Years for 8 Months

Using the formula for \( x \leq 2 \): \[ f\left(\frac{2}{3}\right) = 10.5 \times \frac{2}{3} = 7 \text{ dog-years} \]

Step 4: Calculate Dog-Years for 1 Year and 4 Months

Using the formula for \( x \leq 2 \): \[ f\left(\frac{4}{3}\right) = 10.5 \times \frac{4}{3} = 14 \text{ dog-years} \]

Step 5: Calculate Dog-Years for 4 Years

Using the formula for \( x > 2 \):\[ f(4) = 21 + 4(4-2) = 21 + 8 = 29 \text{ dog-years} \]

Step 6: Calculate Dog-Years for 10 Years

Using the formula for \( x > 2 \):\[ f(10) = 21 + 4(10-2) = 21 + 32 = 53 \text{ dog-years} \]

Step 7: Graph the Function

To graph the function, plot the two pieces based on the input values. For the first interval (\[0 \leq x \leq 2\]), the graph is a line with a slope of 10.5. For the second interval (\[x > 2\]), the graph is a line starting at 21 with a slope of 4, extending indefinitely.

Key Concepts

Human-Years to Dog-Years Conversion

Converting human-years to dog-years is not as simple as a straight seven-to-one ratio. Dogs age and mature much differently than humans do, especially in the early stages of life. This piecewise function provides a better approximation. It treats the first two years of a dog's life as equivalent to 10.5 human-years each. This reflects the rapid maturation period in young dogs when they quickly reach adulthood.
As dogs progress beyond their early years, the second part of the function kicks in. For every human year after the initial two, it uses a conversion rate of four dog-years per human-year. This accounts for the slower, more balanced aging process in adult dogs.
Here's what the conversion formula looks like:

• From 0 to 2 years: \(f(x) = 10.5x\)
• Beyond 2 years: \(f(x) = 21 + 4(x-2)\)

This approach allows for a more nuanced reflection of a dog's aging process, focusing on the significant changes happening in different life stages.

Graphing Functions

Graphing piecewise functions can be initially challenging, but they are incredibly useful for visualizing how a function behaves over different ranges of input values. Think of each segment as a separate linear graph matched to a specific range of inputs.
This particular function displays two pieces:

• The first piece, \(0 \leq x \leq 2\), has a slope of 10.5. Visually, this part of the graph is a straight line rising steeply, illustrating the quick aging pace of young dogs.
• The second piece, for \(x > 2\), starts from the point where the first part stops. The line becomes less steep with a slope of 4, flattening as it stretches out, signifying the slower aging rate of adult dogs.

By graphing, you capture the critical differences in aging rates. You can easily plot this by drawing straight lines with the given slopes, beginning from specific points as dictated by the transitions in the function.
This graphical representation is powerful for interpreting these differences succinctly. It provides an immediate visual comparison of how young and old dogs' ages correlate with human years.

Calculus Application in Real Life

Calculus is not just about abstract formulas and equations; it has practical applications in real life. In this exercise, it helps us understand how dogs grow and mature over time. The concept of slope—a key calculus idea—is crucial here.
The slope in our piecewise function represents the rate at which dogs' age relative to human years. During the first two years, the steep slope of 10.5 indicates rapid growth. Afterward, a gentler slope of 4 shows slower progression. Calculus helps us interpret these changes, as it centers around studying rates and how things change.
Aside from dog ages, calculus is employed in diverse fields:

• Medicine, for analyzing how fast infections spread or how effective treatments are over time.
• Economics, to determine marginal analysis, which assesses costs and benefits.
• Physics, for predicting motion and changes in physical systems.

These real-world applications underscore the importance of calculus, making it a vital tool for solving problems related to growth and change.