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Let's analyze the problem based on each part:

### Given Information:
1. The posts consist of a cuboid base with a right square-based pyramid on top.
2. The width of the cuboid (and hence the width of the pyramid's base) is **20 cm**.
3. The length from the apex \( A \) to a corner of the base (like \( C \)) is **14.6 cm**, accurate to the nearest tenth of a centimeter.

### Problem Breakdown:
- **Part (a):** Find the upper and lower bounds for \( AC \), the length from \( A \) to \( C \).
- **Part (b):** Find the upper and lower bounds for \( AH \), the slant height of the pyramid, where \( H \) is the midpoint of \( BC \).
- **Part (c):** Show that the post is safe by confirming that the angle between the slant height \( AH \) and the base of the pyramid is less than **22°**.

### Solutions:

#### Part (a) Upper and Lower Bounds for \( AC \)
Since \( AC = 14.6 \) cm to the nearest tenth, we can calculate the bounds as follows:
- **Lower bound:** \( 14.6 - 0.05 = 14.55 \) cm
- **Upper bound:** \( 14.6 + 0.05 = 14.65 \) cm

Thus, the possible range for \( AC \) is \( 14.55 \) cm to \( 14.65 \) cm.

#### Part (b) Upper and Lower Bounds for \( AH \)
1. Since \( H \) is the midpoint of \( BC \), the distance \( BH = 10 \) cm (half of the 20 cm base width).
2. We can use the Pythagorean theorem to find \( AH \) as:
   \[
   AH = \sqrt{AC^2 - BH^2}
   \]
3. **Lower bound for \( AH \):**
   \[
   AH_{\text{lower}} = \sqrt{14.55^2 - 10^2} \approx \sqrt{211.7025 - 100} \approx \sqrt{111.7025} \approx 10.57 \, \text{cm}
   \]
4. **Upper bound for \( AH \):**
   \[
   AH_{\text{upper}} = \sqrt{14.65^2 - 10^2} \approx \sqrt{214.6225 - 100} \approx \sqrt{114.6225} \approx 10.71 \, \text{cm}
   \]

So, the range for \( AH \) is approximately **10.57 cm to 10.71 cm**.

#### Part (c) Angle Verification for Safety
1. We are given that for safety, the angle \( \angle BAH \) should be less than 22°.
2. Using trigonometry, \( \tan(\angle BAH) = \frac{BH}{AH} \).
3. Using the lower and upper bounds for \( AH \):
   - For \( AH_{\text{lower}} = 10.57 \):
     \[
     \tan(\angle BAH) = \frac{10}{10.57} \approx 0.946
     \]
     \[
     \angle BAH \approx \tan^{-1}(0.946) \approx 43.5^\circ
    

Vertical posts are to be placed around the outer edge of a children’s park. Each post is formed from a cuboid with a right square-based pyramid on top. The cuboid part of the post is machine-made such that its width, and hence the width of the pyramid, is exactly 20 cm. The length from the apex of the pyramid, A, to any corner of the base of the pyramid is 14.6 cm, but this is only accurate to the nearest tenth of a centimeter. The post is shown in the diagram. (a) Write down the upper bound and lower bound for the possible lengths of edge AC. (b) Determine the upper bound and lower bound for AH, the slant height of the pyramid. (c) Show that this post is safe for children if the angle between the slant height and the base of the pyramid must be less than 22°.

This problem explores the geometry of a cuboid and a right square-based pyramid. Learn how to calculate the bounds for edge AC and slant height AH, and verify the angle between the slant height and the base to ensure the post is safe for children. Suitable for high school students in Grades 10-12.

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