College Algebra, Chapter 2, 2.5, Section 2.5, Problem 30

The period of a pendulum varies directly with the square root of the length of the pendulum.
a.) Express this relationship by writing an equation.
Let $T$ and $\ell$ be the period of length of the pendulum respectively. So, $T = k\sqrt{\ell}$

b.) To double the period, how would we have to change the length $\ell$
Let $T_1$ and $\ell_1$ be the information of the original pendulum. On the other hand, $T_2$ and $\ell_2$ is the information of the new pendulum.

$\begin{equation} \begin{aligned} T_1 &= k \sqrt{\ell_1}\\ \\ k &= \frac{T_1}{\sqrt{\ell_1}} && \Longleftarrow \text{Equation 1}\\ \\ T_2 &= k \sqrt{\ell_2} && \text{Recall that } T_2 = 2T_1\\ \\ 2T_1 &= k \sqrt{\ell_2}\\ \\ k &= \frac{2T_1}{\sqrt{\ell_2}} && \Longleftarrow \text{Equation 2} \end{aligned} \end{equation}$

Use equations 1 and 2 to solve for the requested length.

$\begin{equation} \begin{aligned} \frac{T_1}{\sqrt{\ell_1}} &= \frac{2T_1}{\sqrt{\ell_2}} && \text{Cancel out } T_1\\ \\ \frac{1}{\sqrt{\ell_1}} &= \frac{2}{\sqrt{\ell_2}} && \text{Apply cross multiplication}\\ \\ \sqrt{\ell_2} &= 2 \sqrt{\ell_1} && \text{Square both sides}\\ \\ \ell_2 &= 4 \ell_1 \end{aligned} \end{equation}$

It shows that, in order to double the period, the length must be four times the original length.