# Math Help - Need help Mathematical Models!

1. ## Need help Mathematical Models!

The period (the time for one complete oscillation) of a pendulum is directly proportional to the square root of the length of the pendulum, and a pendulum of length 8 ft has a period of 2 seconds. Find a mathematical model expressing the period of a pendulum as a function of its length:

Why is the answer equal to $P(x) = \sqrt {\frac{x}{2}}$?
and thanks

The period (the time for one complete oscillation) of a pendulum is directly proportional to the square root of the length of the pendulum, and a pendulum of length 8 ft has a period of 2 seconds. Find a mathematical model expressing the period of a pendulum as a function of its length:

Why is the answer equal to $P(x) = \sqrt {\frac{x}{2}}$?
and thanks
I presume x is the length of the pendulum.

We know that P(x) is proportional to $\sqrt{x}$, so let
$P(x) = k \sqrt{x}$ where k is a constant.

Since a pendulum of 8 ft has a period of 2 seconds:
$2 = k \sqrt{8}$

$k = \frac{2}{\sqrt{8}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}}$

Thus
$P(x) = \frac{1}{\sqrt{2}} \sqrt{x} = \sqrt{\frac{x}{2}}$

-Dan