We are given that $$ \frac{d x}{d t}=5 \mathrm{ft} / \mathrm{s} $$ and are asked to find $$ \frac{d \theta}{d t} $$ when $$ x=5 $$. The equation that relates $$ x $$ and $$ \theta $$ can be written from the figure, $$ \frac{x}{12}=\tan (\theta) \quad x=(12 \quad) \tan (\theta) . $$ Differentiating each side with respect to $$ t $$, we get $$ \begin{array}{l} \frac{d x}{d t}=\left(12 \sec ^{2}(\theta)\right. \ \text { so } \ \begin{aligned} \frac{d \theta}{d t} & =(\square) \frac{d \theta}{d t} \ & =(\square) \frac{d x}{d t} \ & =\square) \end{aligned} \end{array} $$ When $$ x=5 $$, the length of the beam is 13 , so $$ \cos (\theta)=\frac{12}{13} $$ and $$ \frac{d 0}{d t}=(\square)\left(\frac{12}{13}\right)^{2}=\square $$

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To solve the problem, we will follow the steps outlined in the prompt.

1. **Extract known conditions:**
   - We have $$ \frac{d x}{d t} = 5 \, \text{ft/s} $$.
   - The relationship between $$ x $$ and $$ \theta $$ is given by $$ x = 12 \tan(\theta) $$.
   - We need to find $$ \frac{d \theta}{d t} $$ when $$ x = 5 $$.

2. **Differentiate the equation:**
   We start with the equation:
   $$
   x = 12 \tan(\theta)
   $$
   Differentiating both sides with respect to $$ t $$:
   $$
   \frac{d x}{d t} = 12 \sec^2(\theta) \frac{d \theta}{d t}
   $$

3. **Rearranging the equation:**
   We can rearrange this to solve for $$ \frac{d \theta}{d t} $$:
   $$
   \frac{d \theta}{d t} = \frac{\frac{d x}{d t}}{12 \sec^2(\theta)}
   $$

4. **Finding $$ \theta $$ when $$ x = 5 $$:**
   When $$ x = 5 $$:
   $$
   5 = 12 \tan(\theta) \implies \tan(\theta) = \frac{5}{12}
   $$
   To find $$ \sec^2(\theta) $$, we use the identity:
   $$
   \sec^2(\theta) = 1 + \tan^2(\theta)
   $$
   Thus,
   $$
   \tan^2(\theta) = \left(\frac{5}{12}\right)^2 = \frac{25}{144}
   $$
   Therefore,
   $$
   \sec^2(\theta) = 1 + \frac{25}{144} = \frac{144 + 25}{144} = \frac{169}{144}
   $$

5. **Substituting values:**
   Now we can substitute $$ \frac{d x}{d t} $$ and $$ \sec^2(\theta) $$ into the equation for $$ \frac{d \theta}{d t} $$:
   $$
   \frac{d \theta}{d t} = \frac{5}{12 \cdot \frac{169}{144}} = \frac{5 \cdot 144}{12 \cdot 169}
   $$
   Simplifying this:
   $$
   \frac{d \theta}{d t} = \frac{5 \cdot 12}{169} = \frac{60}{169}
   $$

6. **Final answer:**
   Thus, when $$ x = 5 $$, we find:
   $$
   \frac{d \theta}{d t} = \frac{60}{169} \, \text{rad/s}
   $$

In summary, the final answer is:
$$
\frac{d \theta}{d t} = \frac{60}{169}
$$
When $$ x = 5 $$, $$ \frac{d \theta}{d t} = \frac{60}{169} \, \text{rad/s} $$.

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