Mathematical model for period of pendulum

Question

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.

Answer 1

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.

Answer 2

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.