answer:You've refactored the code into reusable functions. To handle the case of an empty list and add input validation, you can modify the functions as follows: ```python # Function to find the minimum and maximum numbers using built-in functions def find_min_max_builtin(numbers): if not isinstance(numbers, list) or not all(isinstance(num, (int, float)) for num in numbers): raise ValueError(Input must be a list of numbers) if not numbers: return None, None # or any other default value you prefer return min(numbers), max(numbers) # Function to find the minimum and maximum numbers manually def find_min_max_manual(numbers): if not isinstance(numbers, list) or not all(isinstance(num, (int, float)) for num in numbers): raise ValueError(Input must be a list of numbers) if not numbers: return None, None # or any other default value you prefer min_number = max_number = numbers[0] for num in numbers[1:]: if num < min_number: min_number = num elif num > max_number: max_number = num return min_number, max_number # Example usage: numbers = [4, 2, 9, 6, 5] min_builtin, max_builtin = find_min_max_builtin(numbers) min_manual, max_manual = find_min_max_manual(numbers) print(Minimum (builtin):, min_builtin) print(Maximum (builtin):, max_builtin) print(Minimum (manual):, min_manual) print(Maximum (manual):, max_manual) # Test with empty list numbers = [] min_builtin, max_builtin = find_min_max_builtin(numbers) min_manual, max_manual = find_min_max_manual(numbers) print(Minimum (builtin):, min_builtin) print(Maximum (builtin):, max_builtin) print(Minimum (manual):, min_manual) print(Maximum (manual):, max_manual) # Test with invalid input try: find_min_max_builtin(hello) except ValueError as e: print(Error:, e) try: find_min_max_manual([1, 2, hello, 4]) except ValueError as e: print(Error:, e) ``` In this updated code: * We've added a check at the beginning of each function to ensure the input is a list of numbers. If not, a `ValueError` is raised with a descriptive message. * We've also added a check for an empty list. If the input list is empty, the functions return `(None, None)` as a default value. You can modify this behavior to suit your specific requirements. Note that the input validation uses the `isinstance` function to check if each element in the list is either an `int` or a `float`. The `all` function is used to ensure all elements in the list pass this check.

answer:Here's the Python code to implement Cramer's rule for solving 2 and 3 linear equations: ```python def cramer_rule_2_equations(a, b, c, d, e, f): denominator = (a * d) - (b * c) if denominator == 0: raise ValueError(Denominator is zero. Equations have no unique solution.) x = ((e * d) - (b * f)) / denominator y = ((a * f) - (e * c)) / denominator return x, y def cramer_rule_3_equations(a, b, c, d, e, f, g, h, i, j, k, l): denominator = a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g) if denominator == 0: raise ValueError(Denominator is zero. Equations have no unique solution.) x = (e * (i * l - f * k) - b * (l * h - k * i) + c * (l * g - k * h)) / denominator y = (a * (l * h - k * i) - e * (g * l - d * k) + c * (g * k - d * h)) / denominator z = (a * (e * i - f * h) - b * (d * i - f * g) + e * (d * g - c * h)) / denominator return x, y, z # Example usage for 2 linear equations a, b, c, d, e, f = 2, 1, 4, 3, 7, 11 x, y = cramer_rule_2_equations(a, b, c, d, e, f) print(fSolution for 2 linear equations: x = {x}, y = {y}) # Example usage for 3 linear equations a, b, c, d, e, f, g, h, i, j, k, l = 1, 1, 1, 2, 2, 2, 3, 3, 3, 6, 6, 6 x, y, z = cramer_rule_3_equations(a, b, c, d, e, f, g, h, i, j, k, l) print(fSolution for 3 linear equations: x = {x}, y = {y}, z = {z}) ``` Note that this implementation includes a check for a zero denominator, which indicates that the equations have no unique solution. In such cases, it raises a `ValueError` with a descriptive message.

answer:1. **Identify the dimensions and orientation of the grazing area**: The grazing area is a rectangle with dimensions 45 m and 75 m. To minimize the number of fence posts, we use the longer side (75 m) along the existing rock wall. Thus, the two 45 m sides and one 75 m side need fencing. 2. **Calculate the number of posts for the 75 m side**: Fence posts are placed every 15 m. The number of intervals on the 75 m side is frac{75}{15} = 5. Since a post is needed at both the beginning and end of the fence, the total number of posts for this side is 5 + 1 = 6. 3. **Calculate the number of posts for each 45 m side**: Each 45 m side also has posts every 15 m, giving frac{45}{15} = 3 intervals. Normally, we would need 3 + 1 = 4 posts per 45 m side. However, the posts at the junctions with the rock wall (at 0 m and 75 m) are shared with the 75 m side. Therefore, each 45 m side only needs 4 - 1 = 3 additional posts (subtracting the shared post at the junction). 4. **Calculate the total number of posts required**: Adding the posts from all sides, we have: - 6 posts along the 75 m side. - 3 posts for each of the two 45 m sides, giving 3 times 2 = 6 posts. - Total posts required = 6 + 6 = 12. 5. **Conclusion**: The fewest number of posts required to fence the specified area, using the rock wall as one side, is 12. The final answer is boxed{(B) 12}.

answer:The goal is to merge the two lists of metrics, considering the following conditions: - If the metric name is the same in both lists, then we need to add the values from both lists and update the value in the new list. - If the metric name is not the same in both lists, then we need to keep both metrics in the new list. To accomplish this task, we need to iterate through both lists and compare the metric names. If the metric name is the same in both lists, then we need to add the values from both lists and update the value in the new list. If the metric name is not the same in both lists, then we need to keep both metrics in the new list. Here's how we can implement this: ```python def merge_metrics_v2(list1, list2): assert isinstance(list1, list) and isinstance(list2, list) assert len(list1) > 0 and len(list2) > 0 assert isinstance(list1[0], dict) and isinstance(list2[0], dict) assert 'name' in list1[0] and 'value' in list1[0] assert 'name' in list2[0] and 'value' in list2[0] assert isinstance(list1[0]['name'], str) and isinstance(list2[0]['name'], str) assert isinstance(list1[0]['value'], (int, float)) and isinstance(list2[0]['value'], (int, float)) merged_metrics = [] for metric1 in list1: matching_metric = next((metric2 for metric2 in list2 if metric2['name'] == metric1['name']), None) if matching_metric: merged_metrics.append({'name': metric1['name'], 'value': metric1['value'] + matching_metric['value']}) else: merged_metrics.append(metric1) for metric2 in list2: matching_metric = next((metric1 for metric1 in list1 if metric1['name'] == metric2['name']), None) if not matching_metric: merged_metrics.append(metric2) return merged_metrics ``` In this implementation, we first check if the input is valid by asserting that `list1` and `list2` are both lists of dictionaries with the same schema, and that each dictionary in `list1` and `list2` has two keys, `name` and `value`. We then create a new list to store the merged metrics. Next, we iterate through each metric in `list1` and check if the metric name exists in `list2`. If the metric name exists in both lists, we add the values and update the value in the new list. If the metric name does not exist in `list2`, we add the metric to the new list. We then iterate through each metric in `list2` and check if the metric name exists in `list1`. If the metric name does not exist in `list1`, we add the metric to the new list. Finally, we return the new list of merged metrics.