Solution

Problem

A new way to relate a dog’s age to a human’s age, based on researchers studying labrador retrievers, is discussed at www.caninejournal.com/dog-years-to-human-years. The relationship they found is shown in the graph below.

A description of the graph follows.

A graph entitled Relationship Between Dog Age and Human Age. The horizontal axis is Dog Age (in years) and ranges from $0$ to $14$ . The vertical axis is Human Age (in years) and ranges from $0$ to $100$ . The graph is a curved line that starts at $(0, 0)$ and ends at approximately $(14, 73)$ . The curved line shows a steep increase in Human Age from Dog Age $0$ to $2$ , then a more gradual increase from $2$ to $6$ , and then an even more gradual increase from $6$ to $14$ . The following table provides an approximate data point for each Dog Age on the graph.

Dog Age (years)	Human Age (years)
$0$	$0$
$1$	$31$
$2$	$42$
$3$	$49$
$4$	$53$
$5$	$57$
$6$	$60$
$7$	$62$
$8$	$64$
$9$	$66$
$10$	$68$
$11$	$69$
$12$	$71$
$13$	$72$
$14$	$73$

A traditional way to relate a dog’s age to a human’s age is by multiplying the dog’s age by $7.$ This comparison is called linear because its graph is a straight line. Since multiplication by $7$ implies that $14$ years of dog age equals $7 \times 14 = 98$ years of human age, this line will go from $(0, 0)$ to $(14, 98) .$ Sketch this line carefully on the given graph.

For dog ages of $2$ , $6$ , and $10$ years, use your graph to estimate the human age predicted the traditional way and the new way.

For what dog age are the two predicted human ages farthest apart? About how many years apart are the two predicted human ages?

For what dog ages are the two predicted human ages the same?

If the first year of a cat’s life is equivalent to $15$ human years, the second year to $9$ human years, and each year thereafter to $4$ human years, then show that by the age of $6$ a cat will be younger in human years than either of the predicted dog ages in human years.

Solution

The straight line representing the traditional 'multiply by $7$ ' relationship is shown on the graph.

A straight line connecting the point (0,0) to the point
(14,98) is added to the original graph.

At dog age $2$ , the linear graph predicts about $14$ human years, while the curved graph predicts about $42$ human years.
At dog age $6$ , the linear graph predicts about $42$ human years, while the curved graph predicts about $60$ human years.
At dog age $10$ , the linear graph predicts about $70$ human years, while the curved graph predicts about $68$ human years.

By looking at the vertical distance between the two graphs, the difference between the two predicted human ages for a given dog age appears to be greatest at a dog age of about $2$ years, at which the difference between the two predicted human ages is about $42 - 14 = 28$ years.

The predicted human ages appear to be the same at about $9 \frac{1}{2}$ years and at $0.$

Adding the first six years, the given data predicts that a cat of age $6$ years will compare to a human at $15 + 9 + 4 + 4 + 4 + 4 = 40$ years. Since both dog predictions are greater than that (about $42$ and $60$ ), by the age of $6$ , the equivalent human years for a cat are less than those predicted for a dog.

To Think About: Will a cat remain younger in human years? How do you know?

Problem of the Week Problem B and Solution It’s Been A Dog’s Age

Problem

Solution

Problem of the Week
Problem B and Solution
It’s Been A Dog’s Age