Mathematical model for period of pendulum I'm doing an exercise in calculus about functions as math models, and I'm stuck in this one particular exercise about representing a pendulum's period. The problem goes:
The period of a pendulum is directly proportional to the square root of the length of the pendulum, and a pendulum of length 8 feet has a period of 2 seconds.
So, by definition of "directly proportional to the square root of the length of the pendulum", we have:
$$f(x) = k \sqrt x$$
This next part is where I'm not sure; "a pendulum of length 8 feet has a period of 2 seconds.":
$$2= 8k$$
hence
$$k = 1/4$$
Plugging that in, I arrived with:
$$f(x) = \frac {\sqrt {x} }{4} $$
I took a look at the math model answer and the book says that the correct mathematical model is 
$$f(x) = \sqrt \frac{x}{2} $$
and that if you were to find the period of a pendulum with length 2 feet, it is 1 second.
 A: Your start, with $f(x) = k\sqrt x$, is good. However, when it comes to deciding what $k$ is, you make a mistake. The model says that an $8$ foot pendulum should have a period of $k\sqrt 8$, and being told that this is equal to $2$, you get
$$
2 = k\sqrt8\\
k = \sqrt{\frac12}
$$
This value of $k$ can then be inserted into the general model, and we get
$$
f(x) = \sqrt{\frac12}\cdot \sqrt x = \sqrt{\frac x2}
$$
With this you get $f(2) = 1$, so a pendulum of length $2$ feet does, according to our model, have a period of $1$ second.
A: You are missing "Let $f(x)$ be the period in seconds of a pendulum of length $x$ feet."  Without this sentence, your symbols are not attached to the quantities in the problem.  (We have introduced the symbols in the usual sense of "Sarah, this is Ted; Ted, Sarah.")  You have
$$  f(x) = k \sqrt{x}  \text{.}  $$
Then using the symbols we introduced above, "a pendulum of length $x = 8$ feet has a period of $f(x) = 2$ seconds."  This is
$$  2 = k \sqrt{8}  \text{.} $$
You're using the correct process after this point.
