The problem describes a right triangle formed by the height of a building (225 feet) and its shadow (denoted by $x$ feet). The goal is to find the rate of change of the angle of elevation, $\theta$, as the length of the shadow $x$ changes. Specifically, we are asked to compute $\frac{d\theta}{dx}$ when $x = 264$.

Step-by-Step Solution

1. Set up a trigonometric relationship:

   From the figure, the tangent of the angle $\theta$ is given by: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{225}{x}$

2. Differentiate with respect to $x$:

   To find the rate of change of $\theta$ with respect to $x$, we differentiate both sides of the equation with respect to $x$. Using implicit differentiation, we get: $\frac{d}{dx}[\tan(\theta)] = \frac{d}{dx}\left(\frac{225}{x}\right)$

   The derivative of $\tan(\theta)$ with respect to $x$ is: $\sec^2(\theta) \cdot \frac{d\theta}{dx}$ And the derivative of $\frac{225}{x}$ with respect to $x$ is: $-\frac{225}{x^2}$

   So the equation becomes: $\sec^2(\theta) \cdot \frac{d\theta}{dx} = -\frac{225}{x^2}$

3. Solve for $\frac{d\theta}{dx}$:

   Rearranging to solve for $\frac{d\theta}{dx}$, we have: $\frac{d\theta}{dx} = \frac{-225}{x^2 \cdot \sec^2(\theta)}$

   Since $\sec^2(\theta) = 1 + \tan^2(\theta)$ and $\tan(\theta) = \frac{225}{x}$, we can substitute: $\sec^2(\theta) = 1 + \left(\frac{225}{x}\right)^2 = 1 + \frac{50625}{x^2}$

4. Substitute $x = 264$:

   Now, substitute $x = 264$ into the equation for $\sec^2(\theta)$: $\sec^2(\theta) = 1 + \frac{50625}{264^2} = 1 + \frac{50625}{69696} \approx 1.72639$

   Now substitute this and $x = 264$ into the expression for $\frac{d\theta}{dx}$: $\frac{d\theta}{dx} = \frac{-225}{264^2 \cdot 1.72639} = \frac{-225}{69696 \cdot 1.72639} \approx \frac{-225}{120278.25} \approx -0.001870 \text{ radians per foot}$

Final Answer:

$\theta'(264) \approx -0.00187 \text{ radians per foot}$

Would you like further clarification or additional details?

Here are 5 follow-up questions to enhance your understanding:

1. What is the significance of the negative sign in $\frac{d\theta}{dx}$?
2. How would the rate of change $\frac{d\theta}{dx}$ behave as $x$ increases indefinitely?
3. What is the relationship between $\theta$ and the height of the building in this scenario?
4. How would you approach this problem if the building were on an incline, affecting the angle of elevation?
5. Can this approach be generalized for buildings of different heights?

Tip: In related rates problems involving right triangles, always look for trigonometric relationships (like tangent or sine) to connect the angle with the sides of the triangle.