Gemini 2.5 Flash

The hardest natural substance on Earth is **diamond**. Here's why: * **Mohs Scale of Hardness:** Diamond is rated a 10 on the Mohs scale, which measures scratch resistance. No other natural mineral can scratch a diamond, and diamond can scratch all other natural minerals. * **Atomic Structure:** Diamonds are made of carbon atoms arranged in a very strong, rigid crystal lattice structure, where each carbon atom is bonded to four other carbon atoms in a tetrahedral arrangement. These strong covalent bonds are responsible for its exceptional hardness. While there are synthetic materials that are harder than diamond (like aggregated diamond nanorods or ultrahard fullerite), in terms of naturally occurring substances, diamond holds the top spot.

The capital of Australia is **Canberra**.

800Watts

Let's break down the costs and the increase in value: 1. **Calculate Total Cost:** * Purchase Price: $80,000 * Repair Costs: $50,000 * **Total Investment:** $80,000 + $50,000 = $130,000 2. **Calculate the Increase in Value:** * The value increased by 150% of the original purchase price. * Increase amount = 150% of $80,000 = 1.50 * $80,000 = $120,000 3. **Calculate the Selling Price (New Value):** * Original Value + Increase Amount = Selling Price * $80,000 + $120,000 = $200,000 4. **Calculate the Profit:** * Profit = Selling Price - Total Investment * Profit = $200,000 - $130,000 = $70,000 Josh made a profit of **$70,000**.

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Here's how to calculate Janet's daily earnings: 1. **Eggs used:** 3 (breakfast) + 4 (muffins) = 7 eggs 2. **Eggs remaining to sell:** 16 (laid) - 7 (used) = 9 eggs 3. **Money made:** 9 eggs * $2/egg = $18 Janet makes **$18** every day at the farmers' market.

The capital of Australia is **Canberra**.

Walk

Here's how to break down the profit calculation: 1. **Calculate Sam's total cost:** * He bought a dozen boxes, which is 12 boxes. * Each box cost $10. * Total Cost = 12 boxes * $10/box = $120 2. **Calculate the total number of pens Sam bought:** * 12 boxes * 30 pens/box = 360 pens 3. **Calculate how many packs Sam made:** * He rearranged 360 pens into packs of 6 pens each. * Number of packs = 360 pens / 6 pens/pack = 60 packs 4. **Calculate Sam's total revenue (money made from selling):** * He sold each pack for $5. * Total Revenue = 60 packs * $5/pack = $300 5. **Calculate Sam's total profit:** * Profit = Total Revenue - Total Cost * Profit = $300 - $120 = $180 Sam made a total profit of **$180**.

The time complexity of binary search is **O(log N)** (read as "Big O of log N"). Let's break down why: 1. **The Halving Principle:** Binary search works by repeatedly dividing the search interval in half. * You start with `N` elements. * After the first comparison, you've eliminated half, leaving `N/2` elements to search. * After the second comparison, you've eliminated half of that, leaving `N/4` elements. * This continues until you find the element or the search space is exhausted. 2. **Logarithms:** The number of times you can divide `N` by 2 until you reach 1 (or 0 elements) is precisely what the logarithm base 2 of `N` (`log₂N`) represents. * If `N = 16`, you go: 16 -> 8 -> 4 -> 2 -> 1. That's 4 steps. `log₂16 = 4`. * If `N = 1024`, you go: 1024 -> 512 -> ... -> 1. That's 10 steps. `log₂1024 = 10`. * If `N = 1,000,000`, it's about 20 steps (`log₂1,000,000 ≈ 19.9`). 3. **Efficiency:** This makes binary search incredibly efficient, especially for large datasets, compared to a linear search (which has a time complexity of O(N) because, in the worst case, it might have to check every single element). **In summary:** * **Time Complexity:** O(log N) * **Condition:** The data must be **sorted**. **Space Complexity:** * **Iterative Binary Search:** O(1) - it only uses a few variables (pointers for low, high, mid). * **Recursive Binary Search:** O(log N) - due to the call stack depth, which is proportional to the number of recursive calls (log N).