🔑:The term "mathematics" has undergone significant evolution from its ancient Greek origins to its modern conception, influenced by historical, linguistic, and cultural factors. The word "mathematics" is derived from the Greek term "μάθημα" (máthema), which means "knowledge" or "learning." In ancient Greece, the term referred to the study of knowledge, particularly in the fields of philosophy, astronomy, and geometry.Ancient Greek Origins (6th century BCE - 300 CE)The earliest recorded use of the term "máthema" dates back to the 6th century BCE, in the works of the Greek philosopher Pythagoras. The Pythagoreans, a school of philosophers and mathematicians, used the term to describe their studies in geometry, arithmetic, and astronomy. The Greek mathematician Euclid, in his famous work "Elements" (circa 300 BCE), used the term "μάθημα" to refer to the study of geometry and mathematics.Influence of Ptolemy and the Hellenistic Period (300 BCE - 500 CE)The Greek astronomer and mathematician Ptolemy (circa 90-168 CE) played a significant role in the evolution of the term "mathematics." In his work "Almagest" (circa 150 CE), Ptolemy used the term "μάθημα" to describe the study of astronomy and mathematics. During the Hellenistic period, the term "mathematics" became more widely used to refer to the study of mathematical sciences, including geometry, arithmetic, and astronomy.Medieval and Renaissance Periods (500-1600 CE)During the Middle Ages, the term "mathematics" was adopted into Latin as "mathematica," and its meaning expanded to include the study of arithmetic, geometry, and astronomy. The Latin term "mathematica" was used by scholars such as Boethius (circa 480-524 CE) and Fibonacci (circa 1170-1250 CE) to describe their mathematical works. During the Renaissance, the term "mathematics" was revived in its Greek form, and its meaning continued to expand to include new areas of study, such as algebra and calculus.Modern Conception (1600 CE - present)The modern conception of mathematics as a distinct field of study emerged during the 17th century, particularly with the work of mathematicians such as René Descartes (1596-1650 CE) and Isaac Newton (1643-1727 CE). The term "mathematics" became more widely used to refer to the study of mathematical sciences, including algebra, geometry, calculus, and statistics. The development of modern mathematical disciplines, such as topology, abstract algebra, and number theory, further expanded the scope of mathematics.Key Figures and Their Contributions1. Pythagoras (circa 570-495 BCE): Founded the Pythagorean school, which emphasized the study of mathematics, particularly geometry and arithmetic.2. Euclid (circa 300 BCE): Wrote "Elements," a comprehensive treatise on geometry, which became a foundational text for mathematics.3. Ptolemy (circa 90-168 CE): Contributed to the development of astronomy and mathematics, and used the term "μάθημα" to describe his studies.4. Boethius (circa 480-524 CE): Translated Greek mathematical works into Latin, introducing the term "mathematica" to the Western world.5. Fibonacci (circa 1170-1250 CE): Wrote "Liber Abaci," a influential work on arithmetic and algebra, which helped to popularize the term "mathematica."6. René Descartes (1596-1650 CE): Developed analytic geometry, which laid the foundation for modern mathematics.7. Isaac Newton (1643-1727 CE): Developed calculus, which revolutionized mathematics and physics.Linguistic FactorsThe evolution of the term "mathematics" was influenced by linguistic factors, including:1. Greek and Latin roots: The term "mathematics" is derived from Greek and Latin roots, which were adapted and modified over time.2. Language transmission: The transmission of mathematical knowledge from ancient Greece to the Western world was facilitated by the use of Latin as a lingua franca.3. Terminological developments: The development of new mathematical disciplines and concepts led to the creation of new terms and the modification of existing ones.In conclusion, the evolution of the term "mathematics" from its ancient Greek origins to its modern conception was influenced by a combination of historical, linguistic, and cultural factors. Key figures such as Ptolemy and the Pythagoreans played significant roles in shaping the meaning and scope of the term, while linguistic factors, such as the transmission of knowledge through Latin and the development of new terminology, contributed to its evolution over time.

🔑:To determine if it's possible to lift 100kg by 5 meters with 2,943 Watts of power, we need to calculate the energy required to lift the weight and then see if the given power can achieve this within a reasonable time frame.First, let's calculate the energy required to lift 100kg by 5 meters. The energy required to lift an object is given by the formula:Energy (E) = Work (W) = Force (F) × Distance (d)Since we're lifting the object against gravity, the force required is equal to the weight of the object, which is:Force (F) = mass (m) × acceleration due to gravity (g)= 100 kg × 9.8 m/s²= 980 NNow, we can calculate the energy required:Energy (E) = Force (F) × Distance (d)= 980 N × 5 m= 4900 JSo, 4900 Joules of energy are required to lift 100kg by 5 meters.Next, let's see if the given power of 2,943 Watts can provide this energy. Power is the rate at which energy is transferred, so we can use the formula:Power (P) = Energy (E) / Time (t)Rearranging the formula to solve for time, we get:Time (t) = Energy (E) / Power (P)= 4900 J / 2943 W= approximately 1.67 secondsThis means that, in theory, it is possible to lift 100kg by 5 meters with 2,943 Watts of power, and it would take approximately 1.67 seconds to do so.However, there are a few things to consider:1. Efficiency: The calculation assumes 100% efficiency, which is not possible in real-world scenarios. Some energy will be lost as heat, friction, or other forms of energy dissipation.2. Mechanical advantage: The calculation assumes a direct lift, without any mechanical advantage (e.g., pulleys, levers, or gears). In practice, you might use a system with mechanical advantage to reduce the force required, but this would also increase the distance over which the force is applied.3. Power delivery: The power source might not be able to deliver the full 2,943 Watts instantly. There might be a ramp-up time or limitations in the power delivery system.In conclusion, while the calculation suggests that it's theoretically possible to lift 100kg by 5 meters with 2,943 Watts of power, the actual time it takes might be longer due to efficiency losses, mechanical advantages, and power delivery limitations. Nevertheless, the calculation provides a useful estimate and demonstrates that, in principle, the given power is sufficient to achieve the desired task.

🔑:To determine which investment opportunity is more attractive, we can use the Capital Asset Pricing Model (CAPM) to calculate the expected return for each company based on its beta, the risk-free rate, and the market return, and then compare these expected returns to the actual expected returns provided for each investment.The CAPM formula is:Expected Return = Risk-Free Rate + Beta * (Market Return - Risk-Free Rate)## Step 1: Calculate the expected return for the first investment opportunity using CAPM.For the first company with a beta of 0.8, the expected return according to CAPM is:Expected Return = 4% + 0.8 * (10% - 4%)Expected Return = 4% + 0.8 * 6%Expected Return = 4% + 4.8%Expected Return = 8.8%## Step 2: Calculate the expected return for the second investment opportunity using CAPM.For the second company with a beta of 1.2, the expected return according to CAPM is:Expected Return = 4% + 1.2 * (10% - 4%)Expected Return = 4% + 1.2 * 6%Expected Return = 4% + 7.2%Expected Return = 11.2%## Step 3: Compare the CAPM expected returns with the given expected returns for each investment opportunity.For the first investment, the CAPM expected return is 8.8%, but the actual expected return is 12%. This means the first investment has a higher actual expected return than the CAPM predicted return, indicating it might be undervalued or have an attractive risk-return profile.For the second investment, the CAPM expected return is 11.2%, and the actual expected return is 15%. This also indicates that the second investment has a higher actual expected return than the CAPM predicted return, suggesting it too might be undervalued or have a good risk-return profile.## Step 4: Determine which investment is more attractive based on the comparison.Both investments offer higher expected returns than their respective CAPM predictions, suggesting both could be attractive. However, to choose between them, we consider the excess return over the CAPM prediction. The first investment offers a 3.2% excess return (12% - 8.8%), and the second investment offers a 3.8% excess return (15% - 11.2%). The second investment, therefore, offers a slightly higher excess return over its CAPM predicted return, making it potentially more attractive from a risk-adjusted perspective.The final answer is: boxed{Second investment}

🔑:## Step 1: Calculate the line-to-neutral voltageFirst, we need to calculate the line-to-neutral voltage ((V_{ph})) from the given line-to-line voltage ((V_L)). For a Y-configuration, (V_{ph} = frac{V_L}{sqrt{3}}). Substituting (V_L = 400V), we get (V_{ph} = frac{400}{sqrt{3}}).## Step 2: Compute the line-to-neutral voltage valueCalculate the value of (V_{ph}). (V_{ph} = frac{400}{sqrt{3}} approx frac{400}{1.732} approx 230.94V).## Step 3: Calculate the current in each phaseThe current in each phase ((I_{ph})) can be found using the formula (I_{ph} = frac{V_{ph}}{Z}), where (Z = 10Omega). Substituting (V_{ph}) and (Z), we get (I_{ph} = frac{230.94}{10} approx 23.094A).## Step 4: Calculate the power consumed by one phaseThe power consumed by one phase ((P_{ph})) is given by (P_{ph} = V_{ph} times I_{ph} times pf), where (pf = 0.8). Substituting the values, (P_{ph} = 230.94 times 23.094 times 0.8).## Step 5: Compute the power consumed by one phaseCalculate the value of (P_{ph}). (P_{ph} = 230.94 times 23.094 times 0.8 approx 4239.19W).## Step 6: Calculate the total power consumed by the three-phase loadThe total power ((P_{total})) consumed by the three-phase load is (P_{total} = 3 times P_{ph}), since there are three phases. Substituting (P_{ph}), (P_{total} = 3 times 4239.19).## Step 7: Compute the total power consumedCalculate the total power. (P_{total} = 3 times 4239.19 approx 12717.57W).The final answer is: boxed{12717.57}