This problem involves understanding the normal distribution and using the empirical rule (also known as the 68-95-99.7 rule). The normal distribution is characterized by the mean and the standard deviation, and the empirical rule provides guidelines for how data is distributed within certain intervals from the mean.

Here’s a breakdown:

• Mean (µ) = 41.8 cm (center of the distribution).
• Standard deviation (σ) = 5.3 cm.
• Empirical rule states:
  • 68% of data falls within ±1 standard deviation (µ ± 1σ).
  • 95% of data falls within ±2 standard deviations (µ ± 2σ).
  • 99.7% of data falls within ±3 standard deviations (µ ± 3σ).

Given that:

• V is located at the mean (41.8 cm), which is the peak of the curve.
• U and W are symmetrically placed around the mean, and the shaded region extends between them.

We can calculate the positions of U and W using the empirical rule:

1. For ±1σ (68% of data):

   • V = 41.8 cm, so:
     • U = V - σ = 41.8 - 5.3 = 36.5 cm.
     • W = V + σ = 41.8 + 5.3 = 47.1 cm.
   • The shaded region between U and W would represent approximately 68% of the total area.

2. For ±2σ (95% of data):

   • U = V - 2σ = 41.8 - 2(5.3) = 31.2 cm.
   • W = V + 2σ = 41.8 + 2(5.3) = 52.4 cm.
   • The shaded region would now represent 95% of the area.

Looking at the diagram, the shaded area appears to be the range between U (approximately 30 cm) and W (approximately 55 cm), which corresponds to 95% of the area (as these are roughly ±2σ from the mean).

Thus, the percentage of the total area shaded is likely 95%, and the values of U, V, and W are approximately:

• U = 31.2 cm
• V = 41.8 cm
• W = 52.4 cm

Would you like more details on this or have any further questions?

Here are 5 follow-up questions to deepen understanding:

1. How would the values of U and W change if the standard deviation increased?
2. What is the significance of the empirical rule in analyzing data distributions?
3. How can we interpret the percentages (68%, 95%, 99.7%) in real-world contexts?
4. What would the graph look like if the standard deviation were much smaller?
5. Can you calculate the percentage of area beyond W (greater than 52.4 cm)?

Tip: When analyzing normal distributions, always first identify the mean and standard deviation, as they define the key intervals around which data is distributed.