Poolside: Laguna M.1

The hardest natural substance on Earth is **diamond**, which has a Mohs hardness of 10, the highest on the scale. However, recent studies suggest there are a few other contenders that may surpass diamond under certain conditions: 1. **Wurtzite Boron Nitride (w-BN):** This rare mineral, composed of boron and nitrogen, has a hexagonal structure similar to diamond. Under high pressure and temperature, some experiments indicate it can be harder than diamond, though it is not as widely found in nature. Its natural occurrence is extremely limited and mostly synthesized in labs. 2. **Lonsdaleite:** A hexagonal allotrope of carbon (like diamond's cubic form), lonsdaleite is thought to form under high-pressure conditions during meteorite impacts. While simulations suggest it could be up to 58% harder than diamond, its natural abundance is scarce, and its hardness is disputed due to practical limitations in testing. ### Important Notes: - **Natural Occurrence:** Diamond remains the most widely recognized natural substance in terms of hardness, as the others are rare, primarily synthetic, or have not been fully confirmed in nature. - **Testing Methods:** The Mohs scale is qualitative and limited. More precise tests (e.g., Vickers hardness) show some materials may outperform diamond in specific conditions. - **Toughness vs. Hardness:** Diamond is exceptionally hard but can fracture if struck. Toughness (resistance to breaking) is different, and other materials may excel here. In summary, while **diamond** is still the standard answer due to its natural prevalence and confirmed hardness, emerging materials like wurtzite boron nitride and lonsdaleite challenge this claim under specialized circumstances.

The capital of Australia is **Canberra**. Here's a concise breakdown of why this is the case: 1. **Historical Context**: When Australia federated in 1901, Sydney and Melbourne vied to be the capital. To resolve this, the Constitution mandated a new, neutral capital in a federal territory. 2. **Selection and Development**: Canberra was chosen as the site in 1913, located in the **Australian Capital Territory (ACT)**, midway between Sydney and Melbourne. The city was purposefully designed and constructed to serve as the nation's capital. 3. **Key Institutions**: - The **Parliament House** (opened in 1927) and the **High Court of Australia** are based in Canberra. - It serves as the political and administrative center of the country, distinct from the larger cities like Sydney and Melbourne. This decision avoided giving undue political influence to any existing state, ensuring balance within Australia's federal structure.

3500Watts

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Janet's ducks lay **16 eggs per day**. She uses **3 eggs** for breakfast and **4 eggs** for baking muffins each day. To find the number of eggs she sells: \[ \text{Eggs used} = 3 + 4 = 7 \] \[ \text{Eggs sold} = 16 - 7 = 9 \] Each egg is sold for **$2**, so her daily earnings are: \[ \text{Daily earnings} = 9 \times \$2 = \$18 \] **Answer:** Janet makes $\boxed{18}$ dollars every day at the farmers' market.

The capital of Australia is **Canberra**, located in the **Australian Capital Territory (ACT)**. It was selected as the federal capital in 1913 to resolve a long-standing rivalry between Sydney and Melbourne, the two largest cities at the time. Canberra was purpose-built to house Australia's government institutions, including Parliament House, the High Court of Australia, and the Australian War Memorial. The city's design was influenced by American architect Walter Burley Griffin, who won an international competition for its layout. While Sydney and Melbourne are larger and more internationally recognized, Canberra serves as the political and administrative center of the country.

Walk

Sam bought a dozen boxes (12 boxes) of highlighter pens, each containing 30 pens, at $10 per box. - **Total cost**: \(12 \text{ boxes} \times \$10/\text{box} = \$120\) He rearranged the pens into packs of 6 pens each. - **Total pens**: \(12 \text{ boxes} \times 30 \text{ pens/box} = 360 \text{ pens}\) - **Total packs**: \(360 \text{ pens} \div 6 \text{ pens/pack} = 60 \text{ packs}\) He sold each pack for $5. - **Total revenue**: \(60 \text{ packs} \times \$5/\text{pack} = \$300\) - **Profit**: \(\text{Total revenue} - \text{Total cost} = \$300 - \$120 = \$180\) \[ \boxed{180} \]

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The time complexity of binary search is **O(log n)**. Here's the breakdown: 1. **Mechanism**: Each step reduces the search space by half. After the first comparison, only half the array remains. This halving continues until the target element is found or the search space is empty. 2. **Mathematical Reasoning**: For an array of size \(N\), the number of comparisons needed in the **worst case** (e.g., when the element is absent) is given by how many times \(N\) can be halved until it reaches \(1\). This quantity is \(\log_2(N)\), since \(2^{\log_2(N)} = N\). Thus, the worst-case time complexity is \(O(\log N)\). 3. **Recurrence Relation**: The recurrence relation for binary search is \(T(n) = T(n/2) + O(1)\), which solves to \(O(\log n)\) via logarithmic reduction. This aligns with the **Master Theorem** for recursive algorithms. 4. **Best and Worst Cases**: - **Best-case**: The element is found immediately in the first comparison (\(O(1)\)). - **Worst-case**: The search space is halved until only one element remains. However, time complexity typically refers to the worst-case scenario, so \(O(\log n)\) is the correct answer. 5. **Logarithm Base Irrelevance**: The base of the logarithm (e.g., \(\log_2(N)\) vs. \(\log_{10}(N)\)) is ignored in Big-O notation because constant factors do not affect the growth rate. **Conclusion**: Binary search efficiently narrows the search space exponentially, leading to a logarithmic time complexity of **O(log n)** in the worst case.