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 \[ \frac{d 0}{d t}=(\square)\left(\frac{12}{13}\right)^{2}=\square \]

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: \[ \frac{d \theta}{d t} = \frac{60}{169} \]