# 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.

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.
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.