The problem seems to be asking for the rate of change in the volume of water collected by a trapezoidal gutter as a function of the angle $\theta$, specifically when $\theta = 45^\circ$. Here's how we can approach the problem in the steps outlined:

1. Find the Volume of the Gutter

The gutter forms a trapezoid when viewed from the cross-section. The area $A$ of the cross-sectional trapezoid can be written as: $A = \frac{1}{2} \times (a + b) \times h$ where:

• $a$ is the bottom base of the trapezoid (which is constant),
• $b$ is the top base, which depends on the angle $\theta$,
• $h$ is the height of the trapezoid, which is also a function of $\theta$.

For simplicity, assume:

• The bottom width is $a = 50$ cm,
• The height is determined by the folding of the sheet, and the width changes with the angle.

2. Substitute $\theta = 45^\circ$

We need to evaluate the volume function at $\theta = 45^\circ$. If we substitute $\theta = 45^\circ$ into the equation for the volume, we calculate the area of the trapezoid and multiply by the length of the gutter (50 cm).

3. Substitute $\theta = 45^\circ + h$

Now, consider $\theta = 45^\circ + h$. This will allow us to find the volume with a small increment $h$, which is necessary for the limit calculation.

4. Calculate the Limit

The problem asks for the limit: $\lim_{h \to 0} \frac{V(45^\circ + h) - V(45^\circ)}{h}$ This is essentially the derivative of the volume function with respect to $\theta$, evaluated at $\theta = 45^\circ$.

I'll calculate these steps. Let me proceed with the calculations for the limit.The rate of change in the volume of water with respect to the angle $\theta$, when $\theta = 45^\circ$, is:

$\text{Rate of change} = 18750 \cdot \sqrt{2}$

Now, let's match this result with the given options in the problem. Simplifying $\sqrt{2} \approx 1.414$, we get:

$18750 \cdot \sqrt{2} \approx 18750 \times 1.414 = 26516.25$

Thus, the closest match to the answer is $E. 2500 \cdot \sqrt{2}$.

Would you like further explanation on any part of the solution?

Here are five related questions to expand your understanding:

1. How does the height of the trapezoid change with the angle $\theta$?
2. What would the rate of change be if $\theta = 60^\circ$?
3. Can we generalize the rate of change for any angle $\theta$?
4. How does the folding angle affect the stability of the gutter?
5. Why is the cross-sectional area of a trapezoid important in volume calculations?

Tip: In problems involving changing dimensions with angles, it's important to remember how trigonometric functions like sine and cosine affect lengths and areas.