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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Solve the given systems of equations by using an appropriate algebraic method. 3R = 8 - 51 61 = 8R + 11
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.y = x − 4
Using Kirchhoff’s laws (see the chapter introduction; the equations can be found in most physics textbooks) with the circuit shown in Fig. 5.50, the following equations are found. Find the
Solve the given systems of equations by either method of this section. 30P = 55-Q 19P + 140 + 32 = 0
Solve the given systems of equations by determinants. All numbers are approximate. 6541x 3309x + 4397 y = -7732 - 8755y = 7622
Solve each system of equations to the nearest 0.001 for each variable by using a calculator. 3y = 14x - 9 12x + 23y = 0
Answer the given questions about the determinant to the right.What is the value of the determinant if c = d = 0? a с b d
Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. 3x + y z = -3 - x + y - 3z =
Solve the given systems of equations by using an appropriate algebraic method. 90x110y = 40 30x - 15y = 25
In a laboratory experiment to measure the acceleration of an object, the distances traveled by the object were recorded for three different time intervals. These data led to the following
In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.y = 4/3x + 2
Solve each system of equations to the nearest 0.001 for each variable by using a calculator. 5x = y + 3 4x = 2y - 3
The angle θ between two links of a robot arm is given by θ = at3 + bt2 + ct, where t is the time during an 11.8-s cycle. If θ =19.0° for t = 1.00 s, θ = 30.9° for t = 3.00 s, and θ =19.8° for
Answer the given questions about the determinant to the right.What change in value occurs if both the rows and the columns are interchanged? a с b d
In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.5x − 2y = 40
Solve the given systems of equations by using an appropriate algebraic method. 0.42x0.56y= 1.26 0.98x 1.40y = -0.28
Answer the given questions about the determinant to the right.What is the value of the determinant if a = kb and c = kd ? a с b d
Solve each system of equations to the nearest 0.001 for each variable by using a calculator. 0.75u+ 0.67v = 5.9 2.lu - 3.9v = 4.8
In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution
The angles of a quadrilateral shaped parcel of land with two equal angles (see Fig. 5.52) have measures such that ∠A = ∠C + ∠B and ∠A + 2 ∠B + ∠C = 280°. Find the
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.−2y = 7
Answer the given questions about the determinant to the right.How does the value change if a and c are doubled? a с b d
Solve the given systems of equations by determinants. x + 2y = 5 x + 3y = 7
Solve each system of equations to the nearest 0.001 for each variable by using a calculator. 7R 18V + 13. -1.4R+ 3.6V = 2.6 =
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.24x + 40y = 15
Solve the given systems of equations by determinants. 2x - y = 7 x + y = 2
Solve the given problems by determinants, set up appropriate systems of equations. All numbers are accurate to at least two significant digits.A certain 18-hole golf course has par-3, par-4, and
Find the slope and y-intercept of the line with the given equation and sketch the graph using the slope and y-intercept. A calculator can be used to check your graph.1.5x − 2.4y = 3.0
In order to make the coefficients easier to work with, first multiply each term of the equation or divide each term of the equation by a number selected by inspection. Then proceed with the solution
Solve the right triangles with the given parts or state that there is not enough information to solve. Round off results according to Table 4.1. Refer to Fig. 4.37.B = 12.60°, c = 184.2Data from
Find the values of the trigonometric functions. Round off results according to Table 4.1.cot 41.8°Data from Table 4.1 Angles and Accuracy of Trigonometric Functions Measurements of Angle to
Express the given angles to the nearest minute.−65.4°
Find values of the trigonometric functions of the angle (in standard position) whose terminal side passes through the given points, give answers in exact form.(1,1/2)
Find values of the trigonometric functions of the angle (in standard position) whose terminal side passes through the given points, the coordinates are approximate.
Sketch an appropriate figure, unless the figure is given.Find the angle θ in the taper shown in Fig. 4.57. (The front face is an isosceles trapezoid.) 4.90 cm 1.86 cm 4.50 cm 1 1 I Fig. 4.57
A water channel has the cross section of an isosceles trapezoid. See Fig. 4.76. (a) Show that a formula for the area of the cross section is A= bh + h2cot θ. (b) Find A if b =12.6 ft, h =
The vertical cross section of an attic room in a house is shown in Fig. 4.79. Find the distance d across the floor. 28.3° 1.85 m d Fig. 4.79 7 1
The cross section (a regular trapezoid) of a levee to be built along a river is shown in Fig. 4.78. What is the volume of rock and soil that will be needed for a one-mile length of the levee? 50.0
In tracking an airplane on radar, it is found that the plane is 27.5 km on a direct line from the control tower, with an angle of elevation of 10.3°. What is the altitude of the plane?
A straight emergency chute for an airplane is 16.0 ft long. In being tested, the top of the chute is 8.5 ft above the ground. What angle does the chute make with the ground?
The windshield on an automobile is inclined 42.5° with respect to the horizontal. Assuming that the windshield is flat and rectangular, what is its area if it is 4.80 ft wide and the bottom is 1.50
A person standing on a level plain hears the sound of a plane, looks in the direction of the sound, but the plane is not there (familiar?). When the sound was heard, it was coming from a point at an
A water slide at an amusement park is 85 ft long and is inclined at an angle of 52° with the horizontal. How high is the top of the slide above the water level?
The distance from the ground level to the underside of a cloud is called the ceiling. See Fig. 4.80. A ground observer 950 m from a searchlight aimed vertically notes that the angle of elevation of
The main span of the Mackinac Bridge (see Fig. 4.84) in northern Michigan is 1160 m long. The angle subtended by the span at the eye of an observer in a helicopter is 2.2°. Show that the distance
The window of a house is shaded as shown in Fig. 4.81. What percent of the window is shaded when the angle of elevation θ of the sun is 65°? 2.5 ft 3.2 ft 2.0 ft 0 - Window Fig. 4.81
A Coast Guard boat 2.75 km from a straight beach can travel 100. Find the gear angle θ in Fig. 4.90, if t = 0.180in. at 37.5 km/h. By traveling along a line that is at 69.0° with the beach, how
The impedance Z and resistance R in an AC circuit may be represented by letting the impedance be the hypotenuse of a right triangle and the resistance be the side adjacent to the phase angle ∅. If
In the structural support shown in Fig. 4.83, find x. 31.0° X 21.8° 14.2 in. Fig. 4.83
The surface of a soccer ball consists of 20 regular hexagons (six sides) interlocked around 12 regular pentagons (five sides). See Fig. 4.88. (a) If the side of each hexagon and pentagon
A typical aqueduct built by the Romans dropped on average at an angle of about 0.03° to allow gravity to move the water from the source to the city. For such an aqueduct of 65 km in length, how much
A laser beam is transmitted with a “width” of 0.00200°. What is the diameter of a spot of the beam on an object 52,500 km distant? See Fig. 4.87. 0.00200⁰ Fig. 4.87 52,500 km d
Find the gear angle θ in Fig. 4.90, if t = 0.180 in. t 0.355 in. t Fig. 4.90 t
Each side piece of the trellis shown in Fig. 4.86 makes an angle of 80.0° with the ground. Find the length of each side piece and the area covered by the trellis. -2.25m 2.25m Fig.4.86 80.0⁰
A uniform strip of wood 5.0 cm wide frames a trapezoidal window, as shown in Fig. 4.92. Find the left dimension l of the outside of the frame. 1 22.5° 65.0 cm Fig. 4.92 -5.0 cm
In Example 3, what is the solution if x = −3?Data from Example 3Graph the equation 2x − y − 4 = 0.We first need to find several pairs of numbers (x, y) that are solutions to the equation,
A ground observer sights a weather balloon to the east at an angle of elevation of 15.0° . A second observer 2.35 mi to the east of the first also sights the balloon to the east at an angle of
Make the given changes in Example 1 of this section and then solve the resulting system of equations.In the second equation, change the constant to the right of the = sign from 11 to 12, and in the
A crop-dusting plane flies over a level field at a height of 25 ft. If the dust leaves the plane through a 30° angle and hits the ground after the plane travels 75 ft, how wide a strip is dusted?
Through what angle θ must the crate shown in Fig. 4.89 be tipped in order that its center of gravity C is directly above the pivot point P?
A hang glider is directly above the shore of a lake. An observer on a hill is 375 m along a straight line from the shore. From the observer, the angle of elevation of the hang glider is 42.0°, and
Make the given change in the indicated examples of this section and then solve the resulting problems.In Example 1, interchange the first and second rows of the determinant and then evaluate it.Data
Make the given changes in the indicated examples of this section and then solve the resulting problems.In Example 1, change the + to − in the second equation and then solve the system of
In Example 2, is F1 = 20lb, F2 = 40lb a solution?Data from Example 2Are the forces in the following system given by F1 = 18 lb and F2 = 41lb ?2F1 + 4F2 = 200F2 = 2F1Substituting
A patio is designed in the shape of an isosceles trapezoid with bases 5.0 m and 7.0 m. The other sides are 6.0 m each. Write one or two paragraphs explaining how to use (a) The sine (b) The
Make the given changes in Example 1 of this section and then solve the resulting system of equations.Change the second equation to 8x + 9z = 10 (no y-term).Data from Example 1Solve the following
Determine each of the following as being either true or false. If it is false, explain why.x = 2, y = −3 is a solution of the linear equation 4x − 3y = −1.
In Example 2(a), change the 6 to −6 and then evaluate.Data from Example 2(a) | 46 3 17 4(17)(3)(6) 68 - 18 = 50
Make the given changes in the indicated examples of this section and then solve the resulting problems.In Example 3, change the + to − in the second equation and then solve the system of
In Example 5, if the first y-coordinate is changed to −2, what changes result in the example?Data from Example 5Find the slope of the line through (−1, 2) and (3, −1). In Fig. 5.6, we draw the
In the second equation of Example 4, if − is replaced with +, what is the solution?Data from Example 4Solve the system of equations2x + 5y = 103x - y =6We could write each equation in
Is x = −2, y = 3 a solution to the system of equationsExplain. 2x + 5y = 11 5x = 12? y = -
In Example 3, change the constant to the right of the = sign in the first equation from −1 to −3, change the constant to the right of the = sign in the third equation from 9 to 11, and then solve
Find the slope of the line through (2,−5) and (−1, 4).
Determine each of the following as being either true or false. If it is false, explain why.The slope of the line having intercepts (0,−3) and (2, 0) is 3/2 .
In Example 2(a), change the 4 to −4 and the 6 to −6 and then evaluate.Data from Example 2(a) | 46 3 17 4(17)(3)(6) 68 - 18 = 50
In the first line of Example 8, if + is changed to −, what changes result in the example?Data from Example 8Find the slope and the y-intercept of the line 2x + 3y = 4. We must first write the
In Example 7, what changes occur if the 6 in the first equation is changed to a 2?Data from Example 7Solve the system of equationsx = 2y + 66y = 3x - 6Writing each of these equations in
Solve the given systems of equations. x + y + z = 2 1 [=2-x x + y = 1
Make the given changes in the indicated examples of this section and then solve the resulting problems.In Example 4, change x to 2x in the second equation and then solve the system of equations.Data
Evaluate the given third-order determinants. 5 -2 7 4 -1 3 1 -6
In Example 3, change the + to − in the first equation and then solve the system of equations.Data from Example 3Solve the following system of equations by determinants:First, note that the
Determine each of the following as being either true or false. If it is false, explain why.The system of equations 3x + y = 5, 2y + 6x = 10 is inconsistent.
The base of a 75-ft fire truck ladder is at the rear of the truck and is 4.8 ft above the ground. If the ladder is extended backward at an angle of 62° with the horizontal, how high up on the
For a car rounding a curve, the road should be banked at an angle θ according to the equation tanθ = v2/gr. Here, v is the speed of the car and r is the radius of the curve in the road. See Fig.
A surveyor measures two sides and the included angle of a triangular tract of land to be a = 31.96 m, b = 47.25 m, and C = 64.09°. (a) Show that a formula for the area A of the tract is A =
The apparent power S in an electric circuit in which the power is P and the impedance phase angle is θ is given by S = P secθ. Given P = 12.0 V · A and θ = 29.4°, find S.
For the isosceles triangle shown in Fig. 4.71, show that c = 2asinA/2. a A C Fig. 4.71 a
The area of a quadrilateral with diagonals d1 and d2 is A = 1/2d1d2sinθ, where d1 and d2 are the diagonals and θ is the angle between them. Find the area of an approximately four-sided grass
In Fig. 4.73, find a formula for h in terms of d, α, and β. Fig. 4.73 B h
The voltage e at any instant in a coil of wire that is turning in a magnetic field is given by e = E cosα, where E is the maximum voltage and α is the angle the coil makes with the field. Find the
In Fig. 4.72, find the length c of the chord in terms of r and the angle θ/2.Fig. 4.72 0
A pendulum 1.25 m long swings through an angle of 5.6°. What is the distance between the extreme positions of the pendulum?
A sloped cathedral ceiling is between walls that are 7.50 ft high and 12.0 ft high. If the walls are 15.0 ft apart, at what angle does the ceiling rise?
Find the angle between the line passing through the origin and (3, 2), and the line passing through the origin and (2, 3).
Find the value of x for the triangle shown in Fig. 4.69. 25° 12 Fig. 4.69 X
Explain three ways in which the value of x can be found for the triangle shown in Fig. 4.70. Which of these methods is the easiest? 31° 2 Fig. 4.70 X
Solve the right triangles with the given parts. Refer to Fig. 4.68.A = 49.67°, c = 0.8253 A C b Fig. 4.68 B a C

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