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This question involves calculating percentages based on a normal distribution with the given mean and standard deviation. Let's break down each part.

### Given Data:
- Mean (\(\mu\)) = 49 pounds
- Standard deviation (\(\sigma\)) = 5.7 pounds

For a normal distribution problem, we can use the **z-score formula** to standardize the values:

\[
z = \frac{x - \mu}{\sigma}
\]

Where \(x\) is the value of interest.

### (a) Find the percentage of dogs that weigh less than 49 pounds.
Since 49 pounds is the mean, this is exactly at the center of the normal distribution. By definition, 50% of the data lies below the mean in a normal distribution.

So, **50%** of the dogs weigh less than 49 pounds.

### (b) Find the percentage of dogs that weigh less than 46 pounds.
Here, we need to calculate the z-score for \(x = 46\).

\[
z = \frac{46 - 49}{5.7} = \frac{-3}{5.7} \approx -0.5263
\]

Using the z-table (or a standard normal distribution calculator), the cumulative probability for a z-score of approximately -0.53 is about **0.2981**, which is 29.81%.

Thus, **29.8%** of the dogs weigh less than 46 pounds (rounded to the nearest tenth).

### (c) Find the percentage of dogs that weigh more than 46 pounds.
Since we already know the percentage of dogs that weigh less than 46 pounds, the percentage that weighs more than 46 pounds is simply:

\[
100\% - 29.8\% = 70.2\%
\]

So, **70.2%** of the dogs weigh more than 46 pounds.

### Final answers:
- (a) 50%
- (b) 29.8%
- (c) 70.2%

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How would the percentages change if the standard deviation was smaller?
2. What is the z-score for a dog that weighs 55 pounds?
3. How can you calculate the probability of a dog weighing between 46 and 55 pounds?
4. What percentage of dogs would weigh more than 55 pounds?
5. How would the percentages differ if the mean weight was increased to 52 pounds?

**Tip**: When working with normal distributions, the area under the curve always adds up to 100%, so if you calculate one side, the other is easily found by subtraction.

The weights of a certain dog breed are approximately normally distributed with a mean of 49 pounds, and a standard deviation of 5.7 pounds. Find the percentage of dogs that weigh (a) less than 49 pounds, (b) less than 46 pounds, and (c) more than 46 pounds.

Solve normal distribution problems involving the weights of a dog breed with a mean of 49 pounds and standard deviation of 5.7 pounds. Learn how to calculate percentages for weights below and above specific values using the z-score formula.

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