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New Syllabus Mathematics Book 1 7th Edition Teh Keng Seng, Loh Cheng Yee, Joseph Yeo, Ivy Chow - Solutions

The perimeter of the shaded region is made up of an arc of a semicircle of radius 12 cm and the arcs of two semicircles of diameter 12 cm. Find the perimeter and the area of the shaded region. -12 cm-
The perimeter of a quadrant of a circle is 71.4 cm. Find its area.
Consider the nonunity feedback system is depicted in Figure CP4.11.(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s).(b) For K = 10, 12, and 15, plot the unit step responses.Determine the steady-state errors and the settling times from the plots.For parts (a) and (b), develop a
Consider the closed-loop system is depicted in Figure CP4.10. The controller gain K can be modified to meet the design specifications.(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s).(b) Plot the response of the closed-loop system for K = 5, 10, and 50.(c) When the controller gain
Consider the closed-loop control system shown in Figure CP4.6. The controller gain is K = 2. The nominal value of the plant parameter is a = 1. The nominal value is used for design purposes only, since in reality the value is not precisely known. The objective of our analysis is to investigate the
Consider the closed-loop control system shown in Figure CP4.5. Develop an m-file script to assist in the search for a value of k so that the percent overshoot to a unit step input is approximately P.O. 10%.The script should compute the closed-loop transfer function T(s) = Y(s)/R(s) and generate
Consider a system with the following closed-loop transfer function 1 T(s)= s+2s+2 Obtain the step response, and determine the percent overshoot. What is the steady-state error?
new suspended, mobile, remote-controlled video camera system to bring three-dimensional mobility to professional football is shown in Figure DP4.8(a) [29]. The camera can be moved over the field, as well as up and down. The motor control on each pulley is represented by the system in Figure
Interest in unmanned underwater vehicles(UUVs) has been increasing recently, with a large number of possible applications being considered.These include intelligence-gathering, mine detection, and surveillance applications. Regardless of the intended mission, a strong need exists for reliable and
A hydrobot is under consideration for remote exploration under the ice of Europa, a moon of the giant planet Jupiter. Figure DP4.6(a) shows one artistic version of the mission. The hydrobot is a self-propelled underwater vehicle that would analyze the chemical composition of the water in a search
An op-amp circuit can be used to generate a rapidly exponentially decaying signal. The circuit shown in Figure DP4.5 can generate the signal2 (t) = −e− t, t > 0, 1000 when the input (t) 1 is a unit step input [6]. Select appropriate values for the resistors and capacitor. Assume an ideal
Lasers have been used in eye surgery for many years. They can cut tissue or aid in coagulation [17].The laser allows the ophthalmologist to apply heat to a location in the eye in a controlled manner. Many procedures use the retina as a laser target.The retina is the thin sensory tissue that rests
Consider the system shown in Figure DP4.3.(a) Determine the range of K1 allowable so that the steady state tracking error is ess 1%.(b) Determine a suitable value for K1 and K so that the magnitude of the steady-state error to a wind disturbance Td (t) = 2t mrad/s, 0 t < 5 s, is less than 0.1
The control of the roll angle of an airplane is achieved by using the torque developed by the ailerons. A linear model of the roll control system for a small experimental aircraft is shown in Figure DP4.2, where G(s) = 1 s2+5s+10 The goal is to maintain a small roll angle due to dis- turbances.
A closed-loop speed control system is subjected to a disturbance due to a load, as shown in Figure DP4.1. The desired speed is d (t) = 100 rad/s, and the load disturbance is a unit step input Td (s) = 1/s.Assume that the speed has attained the no-load speed of 100 rad/s and is in steady state.
A capstan drive for a table slide is described in CDP2.1. The position of the slide x is measured with a capacitance gauge, as shown in Figure CDP4.1, which is very linear and accurate. Sketch the model of the feedback system and determine the response of the system when the controller is an
The block diagram of a machine-tool control system is shown in Figure AP4.8.(a) Determine the transfer function T(s) = Y(s)/ R(s).T(s) = Y(s)/R(s).(b) Determine the sensitivity Sb T .(c) Select K and K1 so that the effects of a unit step disturbance are minimized.
A feedback control system with sensor noise and a disturbance input is shown in Figure AP4.7.The goal is to reduce the effects of the noise and the disturbance. Let R(s) = 0.(a) Determine the effect of the disturbance on Y(s).(b) Determine the effect of the noise on Y(s).(c) Choose gains K and K1
An active realization of a proportional integral controller using an operational amplifier is shown in Figure AP4.6.(a) Determine the transfer function G(s) = V2 (s)/V1 (s).G(s) = V2 (s)/V1 (s).(b) Determine the sensitivity of G(s) with respect to the capacitance C.(c) Determine and plot the
A system that controls the mean arterial pressure during anesthesia has been designed and tested [12]. The level of arterial pressure is postulated to be a proxy for depth of anesthesia during surgery. A block diagram of the system is shown in Figure AP4.5, where the impact of surgery is
An armature-controlled DC motor with feedback is shown in Figure AP4.4. Assume that Km = 10, J = 1, and R = 1. Let the tracking error be E(s) = V(s)− Kt(s) .(a) Determine the required gain, K, to restrict the steady-state error to a ramp input to 0.1 (assume that Td (s) = 0 ).(b) For the gain
A machine tool is designed to follow a desired path so that r(t) = (1− t)u(t), where u(t) is the unit step function. The machine tool control system is shown in Figure AP4.3.(a) Determine the steady-state error when R(s) is the desired path as given and Td (s) = 0.(b) Plot the error e(t) for the
The shoulder joint of a robotic arm uses a DC motor with armature control and a set of gears on the output shaft. The model of the system is shown in Figure AP4.2 with a disturbance torque Td (s)which represents the effect of the load. Determine the steady-state error when the desired angle input
A tank level regulator control is shown in Figure AP4.1(a). It is desired to regulate the level H(s) in response to a disturbance change Q3(s). The block diagram shows small variable changes about the equilibrium conditions so that the desired Hd (s) = 0.Determine the equation for the error E(s),
A robot gripper, shown in part (a) of Figure P4.17, is to be controlled so that it closes to an angle by using a DC motor control system, as shown in part (b). The model of the control system is shown in part (c), where Km = 30, R=1, KK = 1, J = 0.1, and b = 1. (a) Determine the response (t) of
The steering control of a modern ship may be represented by the system shown in Figure P4.16[16, 20]. (a) Find the steady-state effect of a constant wind force represented by Td (s) = 1/ s for K = 10 and K = 25. Assume that the rudder input R(s)
Figure P4.15 shows the model of a two-tank system containing a heated liquid, where T0 (s) is the temperature of the fluid flowing into the first tank and T2 (s) is the temperature of the liquid flowing out of the second tank. The system of two tanks has a heater in the first tank with a
A proposed hypersonic plane would climb to 80,000 feet, fly 3800 miles per hour, and cross the Pacific in 2 hours. Control of the aircraft speed could be represented by the model in Figure P4.14.(a) Find the sensitivity of the closed-loop transfer function T(s) to a small change in the parameter
One form of a closed-loop transfer function is G(s)+kG(s) T(s) = G3(5)+kG4(s) (a) Show that ST k(G2G3-GG4) (G3+kG4)(G+kG) (b) Determine the sensitivity of the system shown in Figure P4.13, using the equation verified in part (a).
A robot uses feedback to control the orientation of each joint axis. The load effect varies due to varying load objects and the extended position of the arm. The system will be deflected by the load carried in the gripper. Thus, the system may be represented by Figure P4.7, where the load torque is
An automatic speed control system will be necessary for passenger cars traveling on the automatic highways of the future. A model of a feedback speed control system for a standard vehicle is shown in Figure P4.6. The load disturbance due to a percent grade Td (s) is also shown. The engine gain Ke
A control system has two forward paths, as shown in Figure P4.4. (a) Determine the overall transfer function T(s) = Y(s)/R(s). (b) Calculate the sensitivity, SG T , using Equation (4.16). (c) Does the sensitivity depend on U(s) or M(s)? FIGURE P4.4 Two-path system. M(s) R(s) + Y(s) U(s) Q(s) G(s)
Reconsider the unity feedback system discussed in E4.14. This time select K = 120 and K1 = 10.The closed-loop system is depicted in Figure E4.15.(a) Calculate the steady-state error of the closedloop system due to a unit step input, R(s) = 1/s, with Td (s) = 0. Recall that the tracking error is
Consider the unity feedback system shown in Figure E4.14. The system has two parameters, the controller gain K and the constant K1 in the process.(a) Calculate the sensitivity of the closed-loop transfer function to changes in K1.(b) How would you select a value for K to minimize the effects of
A closed-loop system is used in a high-speed steel rolling mill to control the accuracy of the steel strip thickness. The transfer function for the process shown in Figure E4.13 can be represented as 1 G(s)= s(s+20) Calculate the sensitivity of the closed-loop transfer function to changes in the
In Figure E4.12, consider the closed-loop system with measurement noise N(s), where G(s) = 100 s+100' K G(s) K, and H(s)= s+5 In the following analysis, the tracking error is defined to be E(s) = R(s)- Y(s): (a) Compute the transfer function T(s) = Y(s)/R(s) and determine the steady-state tracking
Consider the closed-loop system in Figure E4.11, where R(s) + G(s) Ta(s) Y(s) H(s) FIGURE E4.11 Closed-loop system with nonunity feedback. (a) Compute the transfer function T(s) = Y(s)/ R(s). (b) Define the tracking error to be E(s) = R(s) Y(s). Compute E(s) and determine the steady- state tracking
E4.10 Consider the feedback control system shown in Figure E4.10. (a) Determine the steady-state error for a step input in terms of the gain, K. (b)Determine the overshoot for the step response for 40 K 400. (c) Plot the overshoot and the steady-state error versus K.
Submersibles with clear plastic hulls have the potential to revolutionize underwater leisure. One small submersible vehicle has a depth-control system as illustrated in Figure E4.9.(a) Determine the closed-loop transfer function T(s) = Y(s)/R(s).(b) Determine the sensitivity 1SK T and S . K T(c)
Four-wheel drive automobiles are popular in regions where winter road conditions are often slippery due to snow and ice. A four-wheel drive vehicle with antilock brakes uses a sensor to keep each wheel rotating to maintain traction. One such system is shown in Figure E4.8. Find the closed-loop
In laser cutting, it is important for the focusing lens to be placed at an angle perpendicular to the laser, so that the laser beam can be focused. An unfocused laser beam causes an elliptical beam shape, a visual example of a steady-state error. Draw the block diagram of an autofocus system, and
A feedback system has the closed-loop transfer function given by T(s)= s + ps+20 s3+ps2+4s+ (1 - p) Compute the sensitivity of the closed-loop trans- fer function to changes in the parameter p, where p>0. Compute the steady-state error to a unit step input as a function of the parameter p.
A unity feedback system has the loop transfer function 10K L(s) G(s)G(s) = s(s+b) Determine the relationship between the steady-state error to a ramp input and the gain K and system parameter b. For what values of K and b can we guar- antee that the magnitude of the steady-state error to a ramp
A magnetic disk drive requires a motor to position a read/write head over tracks of data on a spinning disk, as shown in Figure E4.4. The motor and head may be represented by the transfer function the required K in order to yield a steady-state error of 0.1 mm for a ramp input of 10 cm/s. 10 G(s) =
A robotic arm and camera could be used to pick fruit, as shown in Figure E4.3(a). The camera is used to close the feedback loop to a microcomputer, which controls the arm [8, 9]. The transfer function for the process is K G(s) = ($+5) (a) Calculate the expected steady-state error of the gripper for
A closed-loop system is used to track the sun to obtain maximum power from a photovoltaic array.The tracking system may be represented by a unity feedback control system and 100 Ge(s)G(s) = TS+1' where = 3 s nominally. (a) Calculate the sen- sitivity of this system for a small change in T. (b)
A digital audio system is designed to minimize the effect of disturbances as shown in Figure E4.1. As an approximation, we may represent G(s) = K2 .(a) Calculate the sensitivity of the system due to K2 . (b) Calculate the effect of the disturbance noise Td (s) on Vo (s). (c) What value would you
A process is designed to follow a desired path described by r(t) = (5 − t + 0.5t2 )u(t)where r(t) is the desired response and u(t) is a unit step function. Consider the unity feedback system in Figure 4.32. Compute the steady-state error ( E(s) = R(s)−Y(s)with Td (s) = 0 ) when the loop
Compute the minimal value of K so that the steady-state error due to a unit step disturbance is less than 10%. a. K 1- b. K = b b c. K 10- 10-1 b d. The steady-state error is for any K.
The sensitivity Sb T is: a. S b. ST c. S ST d. S 1 s+ Kb+1 s+1 s+ Kb+1 s+1 s+ Kb +2 S s+ Kb+2
The steady-state tracking error to a unit step input R(s) = 1/s with Td (s) = 0 is: a. ess K K+K2 K2 b. ess K+K c. ess K K(K + K2) K d. ess K+K
The sensitivity STK of the closed-loop system to variations in K is: a. Sk(s) = s+ KK s+ K(K + K2) Ks b. Sk(s)= (s+ K(K+ K2)) c. Sk(s) = s+KK s+ KK d. Sk(s) = K(s+ KK2) (s+ K(K + K2))
The sensitivity ST K1 of the closed-loop system to variations in K1 is: a. Sk(s)= b. S(s)= Ks (s+ K(K+ K2)) 2s s+ K(K+K) S c. Sk(s) = s+ K(K + K2) K(s+ KK2) d. S(s) = (s+K(K+K))
The closed-loop transfer function is: = KK s+ K (K + K) KK s+ K (K + K) KK1 s-K(K+K). KK1 s + KKs+ KK a. T(s)= b. T(s) = c. T(s) = d. T(s) = =
Consider the closed-loop transfer function A + KA T(s)= A3 + KA where A1, A2, A3, and A4 are constants. Compute the sensitivity of the system to varia- tions in the parameter k. a. S = b. S - k(A2 A3-A1A4) (A3 + KA1)(A + KA) k(A2 A3 + AA4) (A3+KA4)(A+KA)
Consider the two systems in Figure 4.33. R(s) K K Y(s) + R(s) K 0.0099 (i) K Y(s) 0.09 0.09 FIGURE 4.33 Two feedback systems with gains K and K2. These systems have the same transfer function when K = K = 100. Which system is most sensitive to variations in the parameter K? Compute the sensitivity
The loop transfer function of the system in Figure 4.32 is 50 Ge(s)G(s) = TS+10 The sensitivity of the closed-loop system to small changes in is: TS a. S(s) = TS +60 b. S(s)= T TS+10 T c. S(s) = TS +60 d. S(s)= TS TS+10
5. An advantage of using feedback is a decreased sensitivity of the system to variations in the parameters of the process. True or False
4. A disturbance is a desired input signal that affects the system output signal. True or False
3. A primary advantage of an open-loop control system is the ability to reduce the system’s sensitivity. True or False
2. The system sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function for a small incremental change. True or False
1. One of the most important characteristics of control systems is their transient response. True or False
The figure shows a closed container of a uniform cross section, which consists of a rectangle ADCB and a quadrant DEC of a circle with centre D. Given that AB = 14cm AD = 9 cm and Al = BH = CG = EF = 20 cm, find(i) the volume,(ii) the total surface area, of the container.
On 5th June 2011, 124 mm of rainfall was recorded over an area of 28 km². If the rainwater falling onto the area was drained through two channels each with a cross-sectional area of 18 m² at a rate of 26.4 m/s, find the time, to the nearest minute, required for the channels to drain off the rain.
A pipe of length 15 cm has an internal radius of 3.8 cm. The thickness of the pipe is 0.8 cm. Find the total surface area of the pipe.
An open rectangular tank of length 32 cm and breadth 28 cm contains water to a depth of 19 cm. 2580 circular metal discs of diameter 23 mm and height 4 mm are dropped into the tank. Find(i) the new height of water in the tank,(ii) the surface area of the tank that is in contact with the water after
A closed steel cylindrical container has a diameter of 186 mm and its height is of its base radius.(i) Find the total surface area of the container, giving your answer in square centimetres.The lid of the container is now removed.The exterior of the container, including the base, is painted
12. An open rectangular tank of length 18 cm and breadth 16 cm contains water to a depth of 13 cm. The water is poured into a cylindrical container of diameter 17 cm. Find(i) the volume of water in the tank,(ii) the height of water in the cylindrical container,(iii) the surface area of the
A pipe of diameter 64 mm discharges water at a rate of 2.05 mm/s into an empty cylindrical tank of diameter 7.6 cm and height 2.3 m. Find the time required to fill the tank, giving your answer correct to the nearest minute.
A pipe of diameter 2.4 cm discharges water at a rate of 2.8 m/s. Find the volume of water discharged in half an hour, giving your answer in litres.
A copper cylindrical rod of diameter 14 cm and length 47 cm is melted and recast into a wire of diameter 8 mm. Find the length of the wire, giving your answer in metres.
The figure shows a metal pipe of length 35 cm. The internal and external diameters of the pipe are 20 mm and 28 mm respectively. Find the volume of metal used in making the pipe, giving your answer in cubic centimetres.
A tank in the shape of a cylinder of diameter 2.4 m and height 6.4 m contains oil to the brim. Find the number of complete cylindrical containers of base radius 8.2 cm and height 28 cm which can be filled by the oil in the tank.
In a toy factory, 200 wooden closed cylinders of diameter 35 mm and height 7 cm have to be painted. What is the total surface area, in cm³, that needs to be painted? (Take a to be 3.142.)
The figure shows a drinking trough in the shape of a half-cylinder. Find its capacity in litres. 84 cm 15 cm
150 litres of water are poured into a cylindrical drum of diameter 48 cm. Find the depth of water in the drum.
The diameter of the base of a cylinder is 0.4 m and its height is of its base radius. Find the volume of the cylinder, giving your answer in litres.
Complete the table for each closed cylinder. (a) Total Diameter Radius Height Volume surface (b) (c) 4 cm (d) 4 m area 14 cm 704 cm' 20 cm 12 320 cm 528 cm 1056 m
For each of the following closed cylinders, find (i) its volume, (ii) its total surface area. (a) 12 cm (b) m (c) 15 mm. 7 cm 4 m 63 mm
The figure shows a section of a steel pipe of length 12 cm. The internal and external radii of the pipe are 2.1 cm and 2.5 cm respectively.(i) Show that the area of the cross section of the pipe is 1.84pi*c * m ^ 2 (ii) Find the internal curved surface area of the pipe.(iii) Hence, find the total
A closed metal cylindrical can has a base radius of 3.5 cm and a height of 10 cm.(i) Find the total surface area of the can.The lid of the can is now removed. The exterior of the container, including the base, is painted purple.(ii) Find the ratio of the area of the can that is painted, to the
A closed metal cylindrical container has a base radius of 5 cm and a height of 12 cm. (i) Calculate the total surface area of the container. The lid of the container is now removed. The exterior of the container, including the base, is painted green. (ii) Express the area of the container that is
A pipe of radius 2.8 cm discharges water at a rate of 3 m/s. Calculate the volume of water discharged per minute, giving your answer in litres.
For each of the following prisms, find (i) its volume, (ii) its total surface area. (a) (b) 5 5 cm 4cm h 15 cm 7 cm 6 cm 9 cm 5cm 6.cm 9 cm
A swimming pool is 50 m long and 25 m wide. It is 1.2 m deep at the shallow end and 2 m deep at the other end. Find 50 m 25 m 1.2 m 2 m 5001 m (i) the volume of water in the pool when it is full, (ii) the area of the pool which is in contact with the water.
The figure shows an empty hall. Without taking into consideration the thickness of the walls and the roof, find the air space in the hall. 23 m 38 m 80 m 21 m
The figure shows a prism standing on a horizontal, rectangular base BCDE. AABC is a vertical cross section of the prism. B Complete the table. En D AB BC CD Area of Volume of AABC prism (a) 3 cm 4 cm 7 cm (b) 9 cm 11 cm 63 cm (c) 15 cm 300 cm 72 000 cm (d) 24,6 cm 7.8 cm 38 376 cm
By first identifying the base, find the volume of each of the following prisms. (a) (b) (c) 46 cm 75 cm 59 cm 120 cm 28 cm 7xm 18 cm 38 cm 16cm 16 cm 10 cm 5 cm 6 cm 3 cm 9 cm
Calculate (i) the volume, (i) the total surface area, of the prism. 6 cm 5cm 4.5 cm 3 cm 4 cm
Calculate (i) the volume, (ii) the total surface area, of the prism. 20 cm 16 cm 12 cm 9 cm
The volume of the prism is 151.2 cm. Find the value of x. 5.6 cm xcm 12 cm
find the volume of the prism. 5 m 4 m 10 m
Three rooms, each in the shape of a cuboid, where PQRS is the floor, are as shown. Cuboid A: PQ=26 m; QR = 1 m; height = 3 m Cuboid B: PQ QR = 5 m; height = 3 m Cuboid C: PQ=6m; QR = 6 m; height = 1.8 m 26 m Room A 3 m R 5 m P 5 m Q Room B 18 m R 6 m 6 m Q Room C 3 m R 1 m
A cuboid of length 12 cm and breadth 9 cm has a total surface area of 426 cm².(6) Find the height of the cuboid.(ii) Hence, find its volume.
The cross section of a drain is a rectangle 30 cm wide. If water 3.5 cm deep flows through the drain at a rate of 22 cm/s, how many litres of water will flow through in one minute?
A trough, in the form of an open rectangular box, is 1.85 m long, 45 cm wide and 28 cm deep externally. If the trough is made of wood 2.5 cm thick, find the volume of wood used to make this trough, giving your answer in m³.
2.85 million cubic metres of earth were required to fill the disused Sin Seng quarry at Rifle Range Road.(i) If each truck could carry a maximum load of 6.25 m³ of earth per trip, how many trips were required to fill the entire quarry?(ii) The cost of transporting each truckload of earth was $55.
The total surface area of a cube is 433.5 cm². Find its volume.
A metal cube has a volume of 64 cm³. It is to be painted on all its faces. Find the total area of the faces that will be coated with paint.
A fish tank measuring 80 cm by 40 cm contains water to a height of 35 cm. Find(i) the volume of water in the tank, giving your answer in litres,(ii) the surface area of the tank that is in contact with the water, giving your answer in m².

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