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(a) Calculate the area of the cross section of the container.

To calculate the area of the cross-section of the container shown in the image, we'll follow these steps:

1. **Identify the Shape of the Cross Section:**
   - The cross-section appears to be a composite shape consisting of a large semicircle and a smaller semicircle subtracted from it.

2. **Extract Dimensions:**
   - The radius of the large semicircle (R) is given as 15 cm.
   - The radius of the smaller semicircle (r) is given as 7 cm.

3. **Calculate the Area of the Large Semicircle:**
   \[
   \text{Area of large semicircle} = \frac{1}{2} \times \pi \times R^2
   \]
   Substituting \(R = 15 \, \text{cm}\):
   \[
   \text{Area of large semicircle} = \frac{1}{2} \times \pi \times 15^2 = \frac{1}{2} \times \pi \times 225 = 112.5\pi \, \text{cm}^2
   \]

4. **Calculate the Area of the Smaller Semicircle:**
   \[
   \text{Area of smaller semicircle} = \frac{1}{2} \times \pi \times r^2
   \]
   Substituting \(r = 7 \, \text{cm}\):
   \[
   \text{Area of smaller semicircle} = \frac{1}{2} \times \pi \times 7^2 = \frac{1}{2} \times \pi \times 49 = 24.5\pi \, \text{cm}^2
   \]

5. **Calculate the Area of the Cross Section:**
   - The cross-sectional area is the area of the large semicircle minus the area of the smaller semicircle:
   \[
   \text{Area of cross section} = 112.5\pi - 24.5\pi = 88\pi \, \text{cm}^2
   \]

6. **Final Answer:**
   \[
   \text{Area of the cross section} = 88\pi \, \text{cm}^2 \approx 276.46 \, \text{cm}^2
   \]

Would you like further details or have any questions about the calculation? 

Here are some related questions you might find useful:
1. How do you calculate the perimeter of the same cross-section?
2. How does the method change if the shapes are not semicircles but segments of circles?
3. Can this method be extended to find the volume of the entire container?
4. What happens to the cross-sectional area if the dimensions are doubled?
5. How can you use integration to find the area of more complex shapes?

**Tip:** When dealing with composite shapes, always break them down into simpler shapes, calculate each area separately, and then combine them using addition or subtraction depending on the shape's structure.

Calculate the area of the cross section of the container.

Learn how to calculate the area of a cross-sectional composite shape, involving a large and small semicircle, with a detailed step-by-step solution. Suitable for high school students studying geometry and composite areas.

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