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You have the A x B SRAM chip. Calculate the following: (a) The required address bus lines and input/output data bus lines in this SRAM chip. (b) The total number of bits stored in this SRAM chip. (c) The SRAM chip with A×B has one chip-select input (CE#), two control signals for read-memory operation (OE#) and write-memory operation (WE#). This chip operates from a 3.3 V power supply. How many pins are needed for this integrated circuit package? (d) Draw a block diagram and label all input/output terminals in this SRAM chip. (e) Using the SRAM chip with A×B, and an inverter, build the memory unit with 2 A×B.

You have the A x B SRAM chip. Calculate the following: (a) The required address bus lines and input/output data bus lines in this SRAM chip. (b) The total number of bits stored in this SRAM chip. (c) The SRAM chip with A×B has one chip-select input (CE#), two control signals for read-memory operation (OE#) and write-memory operation (WE#). This chip operates from a 3.3 V power supply. How many pins are needed for this integrated circuit package? (d) Draw a block diagram and label all input/output terminals in this SRAM chip. (e) Using the SRAM chip with A×B, and an inverter, build the memory unit with 2 A×B.

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You have the A x B SRAM chip. Calculate the following: (a) The required address bus lines and input/output data bus lines in this SRAM chip. (b) The total number of bits stored in this SRAM chip. (c) The SRAM chip with A × B has one chip-select input (CE#), two control signals for read-memory operation (OE#) and write-memory operation (WE#). This chip operates from a 3.3 V power supply. How many pins are needed for this integrated circuit package? (d) Draw a block diagram and label all input/output terminals in this SRAM chip. (e) Using the SRAM chip with A × B , and an inverter, build the memory unit with 2 A × B .

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[25%] In the following Domino logic circuit, VDD = 2 V. There are parasitic capacitances of 10 fF, 10 fF, 40 fF, 20 fF and 60 fF at nodes 1, 2, 3, 4 and Y, respectively. All the NMOS transistors have an ON -resistance of 10 kΩ and a threshold voltage of 0.43 V. (a) Express X and Y as Boolean functions of inputs A, B, C, D, and E. (b) The CMOS inverter between the two dynamic gates is essential. Why? (c) How would you set the relative driving strength of the NMOS and PMOS in this inverter? Explain. (d) Consider A = B = 1 and C changing from 0 to 1 during the evaluation phase (clk is high). Estimate the propagation delay from C to node 3 using the Elmore model. (e) The above inputs come together with E = D = 0, and node 4 is initially discharged. What is the voltage at Y at the end of the evaluation phase?
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Grading for Numeric Answers Tolerance Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7. In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the particular problem. A number line is marked at 100, 105, and 110, where 105 is the properly-rounded answer. A green shaded region extends beyond 105 by 2 percent of this value in each direction. Significant Figures While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to follow the general rules for significant figures, as the following example illustrates. Suppose you are asked to find the area of a rectangle that is 3.1 cm wide and 4.4 cm long. The full-precision answer from your calculator would be 13.64 cm2, although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 cm2. If you were to enter 13.64 cm2, it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 cm2. Instead, you would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not be deducted any points. Part A What is the area of a rectangle that is 3.1 cm wide and 4.4 cm long? Enter the full-precision answer first to see the corresponding feedback before entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box). Part B Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.) Part C The momentum of an object is determined to be 7.2×10-3 kg⋅m/s. Express this quantity as provided or use any equivalent unit. (Note: 1 kg = 1000 g). Part D Enter the following expression in the answer box below: √2gλ3m, where λ is the lowercase Greek letter lambda.Grading for Numeric Answers Tolerance Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7. In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the particular problem. A number line is marked at 100, 105, and 110, where 105 is the properly-rounded answer. A green shaded region extends beyond 105 by 2 percent of this value in each direction. Significant Figures While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to follow the general rules for significant figures, as the following example illustrates. Suppose you are asked to find the area of a rectangle that is 3.1 cm wide and 4.4 cm long. The full-precision answer from your calculator would be 13.64 cm2, although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 cm2. If you were to enter 13.64 cm2, it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 cm2. Instead, you would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not be deducted any points. Part A What is the area of a rectangle that is 3.1 cm wide and 4.4 cm long? Enter the full-precision answer first to see the corresponding feedback before entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box). Part B Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.) Part C The momentum of an object is determined to be 7.2×10-3 kg⋅m/s. Express this quantity as provided or use any equivalent unit. (Note: 1 kg = 1000 g). Part D Enter the following expression in the answer box below: √2gλ3m, where λ is the lowercase Greek letter lambda.Grading for Numeric Answers Tolerance Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7. In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the particular problem. A number line is marked at 100, 105, and 110, where 105 is the properly-rounded answer. A green shaded region extends beyond 105 by 2 percent of this value in each direction. Significant Figures While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to follow the general rules for significant figures, as the following example illustrates. Suppose you are asked to find the area of a rectangle that is 3.1 cm wide and 4.4 cm long. The full-precision answer from your calculator would be 13.64 cm2, although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 cm2. If you were to enter 13.64 cm2, it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 cm2. Instead, you would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not be deducted any points. Part A What is the area of a rectangle that is 3.1 cm wide and 4.4 cm long? Enter the full-precision answer first to see the corresponding feedback before entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box). Part B Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.) Part C The momentum of an object is determined to be 7.2×10-3 kg⋅m/s. Express this quantity as provided or use any equivalent unit. (Note: 1 kg = 1000 g). Part D Enter the following expression in the answer box below: √2gλ3m, where λ is the lowercase Greek letter lambda.Grading for Numeric Answers Tolerance Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7. In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the particular problem. A number line is marked at 100, 105, and 110, where 105 is the properly-rounded answer. A green shaded region extends beyond 105 by 2 percent of this value in each direction. Significant Figures While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to follow the general rules for significant figures, as the following example illustrates. Suppose you are asked to find the area of a rectangle that is 3.1 cm wide and 4.4 cm long. The full-precision answer from your calculator would be 13.64 cm2, although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 cm2. If you were to enter 13.64 cm2, it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 cm2. Instead, you would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not be deducted any points. Part A What is the area of a rectangle that is 3.1 cm wide and 4.4 cm long? Enter the full-precision answer first to see the corresponding feedback before entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box). Part B Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.) Part C The momentum of an object is determined to be 7.2×10-3 kg⋅m/s. Express this quantity as provided or use any equivalent unit. (Note: 1 kg = 1000 g). Part D Enter the following expression in the answer box below: √2gλ3m, where λ is the lowercase Greek letter lambda.Grading for Numeric Answers Tolerance Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7. In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the particular problem. A number line is marked at 100, 105, and 110, where 105 is the properly-rounded answer. A green shaded region extends beyond 105 by 2 percent of this value in each direction. Significant Figures While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to follow the general rules for significant figures, as the following example illustrates. Suppose you are asked to find the area of a rectangle that is 3.1 cm wide and 4.4 cm long. The full-precision answer from your calculator would be 13.64 cm2, although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 cm2. If you were to enter 13.64 cm2, it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 cm2. Instead, you would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not be deducted any points. Part A What is the area of a rectangle that is 3.1 cm wide and 4.4 cm long? Enter the full-precision answer first to see the corresponding feedback before entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box). Part B Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.) Part C The momentum of an object is determined to be 7.2×10-3 kg⋅m/s. Express this quantity as provided or use any equivalent unit. (Note: 1 kg = 1000 g). Part D Enter the following expression in the answer box below: √2gλ3m, where λ is the lowercase Greek letter lambda.
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5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]
Solution
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