Identifying the constants and variables in statements

Question

Please help me to identify the constants and variables in these statements. Thanks in advance.

Ratio of the circumference of any circle to its diameter.

Height of a boy on a given day.

Height of a boy from 6 years to 12 years of age.

Answer 1

The key to understanding the answer to this question is truly about understanding what exactly constants and variables are.

A ${\bf constant}$ is a number which never changes! Constants are very familiar to us: the natural numbers $0, 1, 2, 3, \ldots$ are all constants, for example.

A ${\bf variable}$ is a quantity (usually associated with some value, like temperature or height, for example) which can change depending on different factors. We often express mathematical relationships using variables to denote how outputs change with respect to different inputs.

With this in mind, let's see if we can't answer your question.

Based on elementary facts from geometry, we know that the circumference of any circle, which we can represent by the variable $c$ is given in terms of the constant $\pi$, which is approximately $3.14159...$ and its diameter, which we can represent by the variable $d$ - specifically, we know the relationship $c = \pi \cdot d$ holds true for ANY circle. Dividing both sides by the diameter, $d$, we see that $\frac{c}{d} = \pi$. The ratio of the circumference to the circle is $\frac{c}{d}$, and since this is always equal to $\pi$, a constant, the ratio is constant!

Next, let's think about the last one: the height of a boy from 6 years to 12 years of age. A lot of factors go into how a person grows, but one thing is true for just about anyone - they grow from age 6 to 12. Hence, since a person's height is not the same from age 6 to 12 (and in fact depends on many different factors), the height of a boy from 6 years to 12 years of age is NOT constant, and so it is variable.

For the height of a boy in a given day, this seems like a dubious question to me. While MOST LIKELY one's height is extremely unlikely to fluctuate much within a given day, it's hard to say whether you can truly call the height of a boy on a given day constant - remember, constant means doesn't change at all, ever! I'm inclined to believe whoever posed this question to you wants you to answer "constant", but if that's the case then you should be skeptical - really, the only way to be absolutely sure that a boy's height is CONSTANT is to look at it at an instant in time!