🔑:## Step 1: Calculate the potential energy of the person before jumpingThe potential energy (PE) of the person before jumping can be calculated using the formula PE = m * g * h, where m is the mass of the person (60 kg), g is the acceleration due to gravity (10 m/s^2), and h is the height of the bridge (12 m). Therefore, PE = 60 kg * 10 m/s^2 * 12 m = 7200 J.## Step 2: Calculate the kinetic energy of the person when the rope is fully extendedSince energy is conserved, the potential energy at the top is converted into elastic potential energy of the spring and kinetic energy of the person at the bottom. However, at the maximum extension of the rope, the person's velocity is momentarily zero, meaning all the energy is stored in the spring. The elastic potential energy (EPE) stored in the spring can be calculated using the formula EPE = 0.5 * k * x^2, where k is the spring constant (20 N/m) and x is the extension of the rope.## Step 3: Determine the extension of the rope when it is fully extendedTo find the extension of the rope, we equate the potential energy at the top to the elastic potential energy at the bottom: m * g * h = 0.5 * k * x^2. Solving for x gives x = sqrt((2 * m * g * h) / k).## Step 4: Calculate the extension of the ropeSubstitute the given values into the equation from step 3: x = sqrt((2 * 60 kg * 10 m/s^2 * 12 m) / (20 N/m)) = sqrt((14400) / 20) = sqrt(720) = 26.83282 m.## Step 5: Calculate the force exerted by the rope on the person at maximum extensionThe force (F) exerted by the rope can be found using Hooke's Law: F = k * x. Substituting the values gives F = 20 N/m * 26.83282 m.## Step 6: Perform the multiplication to find the forceF = 20 N/m * 26.83282 m = 536.6564 N.The final answer is: boxed{537}

🔑:The concept you're referring to is based on the principles of classical mechanics, particularly Newton's laws of motion and the conservation of energy. To understand why acceleration orthogonal to an object's movement direction, such as in the case of a satellite orbiting the Earth, does not require energy, whereas maintaining the same circular pattern using a rocket thruster does, let's delve into the details of both scenarios. Scenario 1: Satellite Orbiting the EarthWhen a satellite orbits the Earth, it is constantly falling towards the Earth due to gravity, but its forward velocity ensures it never gets closer to the Earth. This scenario is an example of uniform circular motion, where the satellite's velocity is constantly changing direction due to the gravitational force acting as a centripetal force. The key points to consider here are:1. Centripetal Force: The gravitational force between the Earth and the satellite provides the centripetal force necessary for the satellite to follow a curved path around the Earth. This force is always directed towards the center of the Earth.2. Conservation of Energy: In a closed system like the Earth-satellite system (ignoring external forces like solar radiation pressure or atmospheric drag), energy is conserved. The satellite's total mechanical energy (kinetic energy + potential energy) remains constant over time. The gravitational potential energy and kinetic energy of the satellite can convert into each other, but the total energy remains constant. This means no external energy input is required to maintain the orbit, as the system's energy is conserved.3. Inertia and Newton's First Law: According to Newton's first law of motion (the law of inertia), an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. In the case of the satellite, the gravitational force is the external force that causes it to deviate from a straight-line path, maintaining its circular orbit. However, the satellite does not require additional energy to change its direction of motion because the force of gravity is providing the necessary centripetal acceleration without any increase in the satellite's speed or energy. Scenario 2: Maintaining a Circular Pattern Using a Rocket ThrusterWhen a spacecraft uses a rocket thruster to maintain a circular pattern, it is applying a continuous force tangentially to its direction of motion to counteract any drag or to adjust its orbit. This scenario involves:1. External Force Application: The rocket thruster applies an external force to the spacecraft. This force is not part of a closed system's natural conservation of energy and inertia, as it requires the expenditure of energy stored in the rocket's fuel.2. Energy Expenditure: Unlike the gravitational force in the satellite's orbit, which does not require energy expenditure to maintain the orbit (since it's a conservative force), the rocket thruster continuously expends energy to produce thrust. This energy is converted from the chemical potential energy stored in the rocket's fuel into kinetic energy of the exhaust gases and an increase in the spacecraft's kinetic and/or potential energy.3. Newton's Third Law: The operation of the rocket thruster is based on Newton's third law of motion, where the action of expelling hot gases out of the back of the rocket results in an equal and opposite reaction force that propels the rocket forward. This process requires the continuous application of force and, consequently, the continuous expenditure of energy. ConclusionIn summary, the key difference between these two scenarios lies in the source and nature of the forces involved. The gravitational force acting on a satellite in orbit is a conservative force that does not require the expenditure of energy to maintain the orbit, as it is part of a closed system where energy is conserved. In contrast, using a rocket thruster to maintain a circular pattern involves the application of a non-conservative force that requires the continuous expenditure of energy stored in the rocket's fuel. The principles of inertia, conservation of energy, and the role of external forces are crucial in understanding why acceleration orthogonal to an object's movement direction in these contexts has different energy requirements.

🔑:## Step 1: Identify the given probability density function (PDF) of X and the transformation to get Y.The PDF of X is given by f(x) = (1/4)e^(-x/4) for x > 0, and Y = 3X + 2.## Step 2: Determine the mean (μ) of X using its PDF.To find the mean of X, we use the formula μ = ∫[0,∞) x*f(x) dx. Substituting f(x) gives μ = ∫[0,∞) x*(1/4)e^(-x/4) dx.## Step 3: Calculate the integral for the mean of X.Using integration by parts with u = x and dv = (1/4)e^(-x/4) dx, we find du = dx and v = -e^(-x/4). The integral becomes μ = [-(1/4)x*e^(-x/4)] from 0 to ∞ + ∫[0,∞) (1/4)e^(-x/4) dx. The first term vanishes at both limits, and the second term simplifies to μ = ∫[0,∞) (1/4)e^(-x/4) dx, which is a standard integral that evaluates to 4.## Step 4: Find the variance of X.The variance of X, Var(X), is given by Var(X) = E(X^2) - [E(X)]^2. We already found E(X) = 4. To find E(X^2), we calculate ∫[0,∞) x^2*(1/4)e^(-x/4) dx.## Step 5: Calculate E(X^2) using integration by parts twice.Let u = x^2 and dv = (1/4)e^(-x/4) dx. Then, du = 2x dx and v = -e^(-x/4). After the first integration by parts, we get E(X^2) = [-(1/4)x^2*e^(-x/4)] from 0 to ∞ + ∫[0,∞) (1/2)x*e^(-x/4) dx. The first term vanishes, and for the second term, we apply integration by parts again with u = x and dv = (1/2)e^(-x/4) dx, yielding du = dx and v = -e^(-x/4). This simplifies to E(X^2) = [-(1/2)x*e^(-x/4)] from 0 to ∞ + ∫[0,∞) (1/2)e^(-x/4) dx. The first term vanishes, and the integral evaluates to E(X^2) = 8.## Step 6: Calculate the variance of X.With E(X) = 4 and E(X^2) = 8*4 = 32 (correcting the calculation from step 5, recognizing the error in evaluating the integral for E(X^2) directly), Var(X) = 32 - 4^2 = 32 - 16 = 16.## Step 7: Find the variance of Y using the properties of variance.Given Y = 3X + 2, Var(Y) = Var(3X + 2) = 3^2 * Var(X) because the variance of a constant times a random variable is the square of the constant times the variance of the variable, and the variance of a constant is 0.## Step 8: Calculate the variance of Y.Var(Y) = 3^2 * Var(X) = 9 * 16 = 144.The final answer is: boxed{144}

🔑:The theory of comparative advantage, first introduced by David Ricardo, suggests that countries should specialize in producing goods for which they have a lower opportunity cost, relative to other countries. Opportunity cost refers to the value of the next best alternative that is given up when a choice is made. In the scenario where a country has an absolute advantage in producing two goods but a comparative advantage in producing one, the country should specialize in the good in which it has a comparative advantage.To illustrate this, let's consider an example. Suppose Country A has an absolute advantage in producing both wheat and cloth, meaning it can produce more of both goods than Country B. However, Country A has a comparative advantage in producing wheat, as it can produce wheat at a lower opportunity cost than Country B. This means that the opportunity cost of producing wheat in Country A is lower than in Country B, even though Country A can produce more cloth than Country B.The concept of scarcity is also relevant here, as it refers to the fundamental economic problem of having unlimited wants and needs in a world with limited resources. In this scenario, both countries face scarcity, as they have limited resources (labor, capital, and land) to devote to producing wheat and cloth. By specializing in the good in which it has a comparative advantage (wheat), Country A can produce more wheat and trade it with Country B for cloth, which Country B can produce more efficiently.The role of self-interest in free market operations is crucial in this scenario. Country A will specialize in producing wheat because it is in its self-interest to do so. By producing wheat at a lower opportunity cost, Country A can maximize its output and trade it with Country B for cloth, which it values more than the cloth it could produce itself. Similarly, Country B will specialize in producing cloth because it is in its self-interest to do so, as it can produce cloth at a lower opportunity cost than wheat.The potential gains from trade in this scenario are significant. By specializing in wheat, Country A can produce more wheat than it needs, and trade the surplus with Country B for cloth. Country B, on the other hand, can produce more cloth than it needs, and trade the surplus with Country A for wheat. Both countries benefit from trade, as they can consume more goods and services than they could if they were self-sufficient.To quantify the gains from trade, let's assume that Country A can produce 100 units of wheat and 50 units of cloth, while Country B can produce 50 units of wheat and 100 units of cloth. If both countries were self-sufficient, they would each produce and consume 50 units of wheat and 50 units of cloth. However, if Country A specializes in producing wheat and Country B specializes in producing cloth, they can produce 100 units of wheat and 100 units of cloth, respectively. By trading 50 units of wheat for 50 units of cloth, both countries can consume 75 units of wheat and 75 units of cloth, which is a significant increase in their standard of living.In conclusion, the theory of comparative advantage applies in this scenario by suggesting that Country A should specialize in producing wheat, in which it has a comparative advantage. The concepts of opportunity cost, scarcity, and self-interest are all relevant, as they drive the decisions of countries to specialize and trade. The potential gains from trade are significant, as both countries can consume more goods and services than they could if they were self-sufficient. By specializing and trading, countries can overcome the limitations imposed by scarcity and maximize their economic well-being.