🔑:## Step 1: Understand the Grey Code SequenceFirst, we need to understand the sequence of a three-bit grey code. A three-bit grey code sequence is as follows: 000, 001, 011, 010, 110, 111, 101, 100. This sequence represents the numbers 0 through 7 in grey code.## Step 2: Determine the Output for Each InputSince the decoder has active low outputs, each output terminal X[0:7] will be low (0) when its corresponding grey code is detected on the inputs A[2:0], and high (1) otherwise.## Step 3: Write Boolean Functions for Each OutputTo write the boolean functions, we need to consider when each output should be low (0). The output will be low for the specific grey code it represents and high for all other combinations.- X[0] should be low for A[2:0] = 000, so its boolean function is (overline{A_2} cdot overline{A_1} cdot overline{A_0}).- X[1] should be low for A[2:0] = 001, so its boolean function is (overline{A_2} cdot overline{A_1} cdot A_0).- X[2] should be low for A[2:0] = 011, so its boolean function is (overline{A_2} cdot A_1 cdot A_0).- X[3] should be low for A[2:0] = 010, so its boolean function is (overline{A_2} cdot A_1 cdot overline{A_0}).- X[4] should be low for A[2:0] = 110, so its boolean function is (A_2 cdot overline{A_1} cdot A_0).- X[5] should be low for A[2:0] = 111, so its boolean function is (A_2 cdot A_1 cdot A_0).- X[6] should be low for A[2:0] = 101, so its boolean function is (A_2 cdot A_1 cdot overline{A_0}).- X[7] should be low for A[2:0] = 100, so its boolean function is (A_2 cdot overline{A_1} cdot overline{A_0}).## Step 4: Consider Active Low AspectSince the outputs are active low, the boolean functions derived directly give us the conditions under which each output is low. To get the actual output (which is the inverse of what we've calculated since we're looking for when it's low), we don't need to invert these functions because they already represent the conditions for the outputs to be low.The final answer is: X[0] = (overline{A_2} cdot overline{A_1} cdot overline{A_0}),X[1] = (overline{A_2} cdot overline{A_1} cdot A_0),X[2] = (overline{A_2} cdot A_1 cdot A_0),X[3] = (overline{A_2} cdot A_1 cdot overline{A_0}),X[4] = (A_2 cdot overline{A_1} cdot A_0),X[5] = (A_2 cdot A_1 cdot A_0),X[6] = (A_2 cdot A_1 cdot overline{A_0}),X[7] = (A_2 cdot overline{A_1} cdot overline{A_0})

🔑:When there is an increase in demand, assuming a fixed supply curve, the equilibrium price and quantity of a good will change. To understand this, let's analyze the situation using a graph.Initial EquilibriumSuppose we have a market for a good with a downward-sloping demand curve (D1) and an upward-sloping supply curve (S). The initial equilibrium is at point E1, where the demand and supply curves intersect. At this point, the equilibrium price is P1 and the equilibrium quantity is Q1.Increase in DemandNow, let's assume that there is an increase in demand, which shifts the demand curve to the right from D1 to D2. This means that at every price level, consumers are now willing to buy more of the good than they were before.New EquilibriumWith the demand curve shifted to the right, the new equilibrium is at point E2, where the demand curve D2 intersects the supply curve S. At this new equilibrium, the equilibrium price has increased to P2, and the equilibrium quantity has increased to Q2.Graphical IllustrationHere's a graph to illustrate the situation:In this graph, the initial demand curve is D1, and the supply curve is S. The initial equilibrium is at point E1 (P1, Q1). When demand increases, the demand curve shifts to the right to D2. The new equilibrium is at point E2 (P2, Q2), where the demand curve D2 intersects the supply curve S.ReasoningThe increase in demand leads to a higher equilibrium price and quantity for several reasons:1. Increased willingness to pay: With an increase in demand, consumers are willing to pay a higher price for the good. This is reflected in the upward shift of the demand curve.2. Shortage at the initial price: At the initial price P1, the quantity demanded (Q2) is greater than the quantity supplied (Q1). This creates a shortage, which drives up the price.3. Supply curve is fixed: Since the supply curve is fixed, the quantity supplied does not change immediately. However, as the price rises, suppliers are incentivized to produce more, which leads to an increase in the equilibrium quantity.In summary, when there is an increase in demand, assuming a fixed supply curve, the equilibrium price and quantity of a good will increase. The demand curve shifts to the right, leading to a higher equilibrium price and quantity, as illustrated in the graph.

🔑:To design a system that meets the specified requirements, we'll explore a combination of materials and a triggering mechanism that allows the liquid to harden rapidly at a chosen moment. The system we propose involves a custom formulation of a polyurethane resin that is opaque, glossy white, and non-adhesive, with a density slightly higher than water. The hardening process will be initiated by a chemical reaction triggered by a catalyst. System Components1. Base Resin: The base of our system will be a specially formulated polyurethane resin. Polyurethane resins can be formulated to have a wide range of properties, including being opaque, glossy, and non-adhesive. By adjusting the resin's composition, we can achieve the desired color (glossy white) and ensure it is non-adhesive after hardening.2. Catalyst: To initiate the hardening process, a catalyst will be added to the resin. This catalyst will react with another component (hardener) to initiate the polymerization process, leading to the hardening of the resin. The choice of catalyst and hardener will be critical in controlling the reaction time and ensuring that the hardening occurs rapidly once triggered.3. Hardener: The hardener will be mixed with the base resin in a specific ratio to ensure complete polymerization when the catalyst is added. The hardener and catalyst will be designed to react rapidly, allowing the resin to harden within a few seconds to minutes after mixing.4. Density Modifier: To achieve a density slightly higher than water, density modifiers such as silica fillers or calcium carbonate can be added to the resin. These fillers not only adjust the density but can also enhance the mechanical properties of the hardened material.5. Triggering Mechanism: For the system to harden at a chosen moment, a controlled mixing system will be designed. This could involve a dual-chamber container where the resin and hardener are stored separately. At the chosen moment, the contents of the two chambers are mixed, either manually or through an automated system, initiating the hardening process. Technical Feasibility- Material Formulation: Formulating a polyurethane resin that meets all the specified properties (opaque, glossy white, non-adhesive, and slightly denser than water) is technically feasible with current materials science capabilities. Adjusting the resin's composition and adding appropriate fillers can achieve the desired properties.- Rapid Hardening: Achieving rapid hardening within a few seconds to minutes is also feasible through the selection of appropriate catalysts and hardeners. The reaction kinetics can be tuned by adjusting the concentration of the catalyst and the hardener, as well as the temperature of the mixture.- Volume Handling: Handling volumes between 10cm and 30cm in diameter is manageable with appropriately sized mixing and application equipment. For larger volumes, scaling up the mixing and application process while maintaining uniform mixing and temperature control will be essential. Potential Limitations- Uniformity and Mixing: Ensuring uniform mixing of the resin and hardener, especially for larger volumes, could be a challenge. Inadequate mixing can lead to variations in hardness, density, and other properties across the hardened material.- Temperature Sensitivity: The reaction rate of the polymerization process can be highly temperature-sensitive. Controlling the temperature of the mixture may be necessary to achieve consistent hardening times.- Shelf Life and Stability: The stability and shelf life of the resin and hardener components must be considered. Some formulations may have limited shelf lives or require special storage conditions to maintain their potency and consistency.- Safety and Environmental Impact: The safety of the materials and the environmental impact of the production process and disposal of the hardened material must be evaluated. Ensuring that the materials are non-toxic and can be disposed of safely is crucial.In conclusion, designing a system for a liquid that hardens rapidly at a chosen moment, with the specified properties, is technically feasible with current technology. However, careful consideration of the material formulation, mixing process, and triggering mechanism, along with addressing potential limitations, is necessary to ensure the system meets all the requirements reliably and safely.

🔑:_Solution_The bandwidth of the music signal is 10 kHz (-) 50 Hz (=) 9.95 kHz, and the highest frequency component of the signal is 10 kHz. The Nyquist rate is twice this value, or 20 kHz. The sampling rate is 1.2 times the Nyquist rate, or 24 kHz. The ADC produces 12 bits per sample, and the sampling rate is 24 kHz. Therefore, the link serial bit rate is 12 (times) 24 (=) 288 kbps. The sampling rate at the receiver must be the same as at the transmitter, or 24 kHz. Therefore, the maximum frequency component of the signal passed by the filter must be one-half of the sampling rate, or 12 kHz.