Shrinking

Task

When working on a report for class, Catrina read that a woman over the age of 40 can lose approximately 0.06 centimeters of height per year.

1. Catrina's aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54 centimeters.)

2. Catrina's 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother's height has also decreased at this rate, about how tall was she at age 40? Explain your reasoning.

IM Commentary

Note that there is a significant amount of rounding in the final answer. This is because people almost never report their heights more precisely than the closest half-inch. If we assume that the heights reported in the task stem are rounded to the nearest half-inch, then we should report the heights given in the solution at the same level of precision.

Solution

If a person loses an average of 0.06 cm per year after age 40 and 1 inch = 2.54 cm, after age 40 they lose, on average $$0.06 \div 2.54 = 0.024 \text{ inches per year.}$$

1. In the 25 years from age 40 to age 65, Nancy could be expected to lose approximately $25 \times 0.024 = 0.6$ inches. Subtracting this from Nancy's current height, Nancy's height at age 65 could be expected to be approximately 5 feet, 6$\frac12$ inches.

2. In the 50 years from age 40 to age 90, Catrina's grandmother could be expected to lose approximately twice Nancy's loss in height, or 1.2 inches. Adding this to Catrina's grandmother's current height, Catrina's grandmother could be expected to have been approximately 5 feet, 2 inches tall at age 40.