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In this problem, we are given a semicircle with area \( A(\theta) \) and a triangle with area \( B(\theta) \), and we are asked to compute the limit:

\[
\lim_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)}
\]

### Step-by-Step Solution:

#### 1. **Area of the Semicircle, \( A(\theta) \):**
   The semicircle has a radius of 10 cm. The area of a full circle is given by:

   \[
   A_{\text{circle}} = \pi r^2 = \pi (10)^2 = 100\pi \text{ square cm}.
   \]

   Since \( A(\theta) \) is the area of the semicircle sector subtended by angle \( \theta \), the formula for the area of a sector is:

   \[
   A(\theta) = \frac{1}{2} \times 100\pi \times \frac{\theta}{2\pi} = 50\theta \text{ square cm}.
   \]

#### 2. **Area of the Triangle, \( B(\theta) \):**
   The triangle \( PQR \) is an isosceles triangle with sides 10 cm. The height of this triangle, corresponding to the line from \( R \) to the midpoint of \( PQ \), is proportional to \( \theta \). For small angles \( \theta \), we approximate the area of the triangle by the small angle approximation for sine:

   \[
   B(\theta) \approx \frac{1}{2} \times 10 \times 10 \times \theta = 50\theta \text{ square cm}.
   \]

#### 3. **Calculating the Limit:**
   Now, we compute the limit:

   \[
   \lim_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} = \lim_{\theta \to 0^+} \frac{50\theta}{50\theta} = 1.
   \]

Thus, the value of the limit is:

\[
\boxed{1}.
\]

Let me know if you'd like more details or have any further questions!

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Here are five related questions you might explore:

1. How does the area of a sector change as the radius increases?
2. What is the relationship between a circle’s circumference and its radius?
3. How can you compute the area of a triangle if it’s not isosceles?
4. What is the approximation of \( \sin(\theta) \) for small angles?
5. How would the limit change if the triangle were not isosceles?

**Tip**: When working with limits involving geometric shapes, small angle approximations like \( \sin(\theta) \approx \theta \) are often very helpful.

If A(θ) is the area of the semicircle and B(θ) is the area of the triangle, find lim θ → 0⁺ A(θ)/B(θ).

This problem involves calculating the limit of the area ratio between a semicircle and a triangle as the angle tends to zero. Using small angle approximation and basic geometry, the areas of both shapes are derived and the limit is evaluated. Suitable for students in Grades 11-12.

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