Mathematics can impact us alot. I was reading an article the other day entitled, "Breed and sex differences in growth curves for two breeds of dog guides," by Helmink, et.al., Journal of Animal Science 2000.78:27-32. The article shares the researchers mathematical modeling of the growth and weight gain of puppies of two breeds of guide dogs, noting that guide dogs need to have sufficient size and weight to guide their owners, but not too heavy or they are too large and difficult for their owners to handle. They state that a weight of 40 to 70 lbs is ideal. A model has been created that will predict weight and hence size very early in the puppy's life, so that costly training to be a guide dog will not be wasted on dogs that will become too large or too small for the work. Since we have weighed our new German Shephard puppy, Zeus, relatively often and had the data available, I decided to plot the data and see if it followed the model that Helmink et al. created. Guess what...it did. The math is below!!!
However , Zeus the Mighty Dog is too big to be a guide dog. See Zeus below...then and now!
The Math...it says what the graph shows to the right!!!. The following function is called a Gompertz function which can be used to model growth. The Gompertz function is a non-linear, sigmoidal function with its point of inflection at 36.8% of the mature weight. This function was used to give a mathematical modeling representation of Zeus's weight gain as he grew.
W(t) = Wmax exp{-e[-(t-c)/b]} is the general Gompertz function where:
W(t) is Zeus's weight at time t in days,
Wmax is Zeus's estimated mature body weight, 95 lbs
b is proportional to Zeus's duration of growth, 74
c is the age at the point of inflection of the curve, 98 days
t is Zeus's age in days.
To fit Zeus's measured data, I adjusted the variables Wmax, b and c to have the curve fit Zeus's weight vs age data. The fitted
equation that describes Zeus's weight gain is : W(t) = 95 exp{-e[-(t-98)/74]}
From the article it says that a dog will continue to grow for 4b+c days, or for Zeus, 394 days. So Zeus should be completely grown about 2 months from now!
Zeus at 2 months Zeus at 10 months
|