Gemini 2.5 Flash-Lite

The hardest natural substance on Earth is **diamond**. It's renowned for its extreme hardness, ranking a perfect 10 on the Mohs scale of mineral hardness. This means it can scratch any other naturally occurring mineral.

The capital of Australia is **Canberra**.

For plug-in solar regulations in the UK, the upcoming changes are focused on allowing systems up to **3.68kW (kilowatts)** to be installed without needing a full grid connection application in many cases. Therefore, the answer is: **3680Watts**.

Here's how to calculate Josh's profit: **1. Calculate the total cost:** * Purchase price: $80,000 * Repair costs: $50,000 * Total cost: $80,000 + $50,000 = $130,000 **2. Calculate the increase in value:** * The repairs increased the value by 150% of the original purchase price. * Increase in value: 150% of $80,000 = 1.50 * $80,000 = $120,000 **3. Calculate the new value of the house:** * Original purchase price: $80,000 * Increase in value: $120,000 * New value: $80,000 + $120,000 = $200,000 **4. Calculate the profit:** * New value of the house: $200,000 * Total cost: $130,000 * Profit: $200,000 - $130,000 = $70,000 **Answer:** Josh made a profit of $70,000.

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Here's how to solve the problem step-by-step: 1. **Eggs laid per day:** Janet's ducks lay 16 eggs. 2. **Eggs eaten for breakfast:** Janet eats 3 eggs. 3. **Eggs used for muffins:** Janet uses 4 eggs for muffins. 4. **Total eggs used:** 3 (breakfast) + 4 (muffins) = 7 eggs 5. **Eggs remaining for sale:** 16 (total) - 7 (used) = 9 eggs 6. **Earnings per egg:** Janet sells each egg for $2. 7. **Total earnings from the farmers' market:** 9 eggs * $2/egg = $18 **Answer:** Janet makes $18 every day at the farmers' market.

The capital of Australia is **Canberra**.

Walk

Here's how to solve the problem step-by-step: **1. Calculate the total number of pens:** * Sam bought 12 boxes * 30 pens/box = 360 pens **2. Calculate the total cost:** * Sam spent 12 boxes * $10/box = $120 **3. Calculate the number of packs he made:** * He made 360 pens / 6 pens/pack = 60 packs **4. Calculate the total revenue from selling the packs:** * He sold 60 packs * $5/pack = $300 **5. Calculate the total profit:** * Profit = Total Revenue - Total Cost * Profit = $300 - $120 = $180 **Answer:** Sam made a profit of $180 in total.

The time complexity of binary search is **O(log n)**, where 'n' is the number of elements in the sorted array. Here's why: * **Logarithmic Nature:** Binary search works by repeatedly dividing the search interval in half. In each step, it eliminates half of the remaining elements. This "halving" is characteristic of logarithmic behavior. * **Worst-Case Scenario:** In the worst-case scenario, you might have to perform `log₂n` divisions to narrow down the search to a single element or to determine that the element is not present. **Let's illustrate with an example:** If you have an array of 16 elements: 1. You check the middle element. You've eliminated 8 elements. 2. You check the middle of the remaining 8. You've eliminated 4 more. 3. You check the middle of the remaining 4. You've eliminated 2 more. 4. You check the middle of the remaining 2. You've eliminated 1 more. You've performed 4 comparisons. And `log₂16 = 4`. **In summary:** * **Best Case:** O(1) - If the element is found at the first comparison (the middle element). * **Average Case:** O(log n) * **Worst Case:** O(log n) Since we usually express time complexity in terms of the worst-case scenario, the answer is consistently **O(log n)**. **Important Prerequisite:** Binary search **requires the input array to be sorted**. If the array is not sorted, you would first need to sort it, which would add its own time complexity (e.g., O(n log n) for efficient sorting algorithms). However, the binary search algorithm itself, on a sorted array, is O(log n).