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study help
mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Find the rectangular equation of each of the given polar equations. r 4 2 - cose
Find the rectangular equation of each of the given polar equations.r2 = 9sinθ
When a circle is graphed on a calculator, there are sometimes gaps in the graph on the left and right sides of the circle. Why does this happen? Does the graph really have gaps?
A rocket is fired horizontally from a plane. Its horizontal distance x and vertical distance y from the point at which it was fired are given by x = v0t and y = 1/2 gt2, where v0 is the initial velocity of the rocket, t is the time, and g is the acceleration due to gravity. Express y as a function
Find the equation of the line that has one negative intercept, passes through (3, 2), and forms with an axis and the line y = 2 a triangle with an area of 12.
The joint between two links of a robot arm moves in an elliptical path (in cm), given bySketch the path. r 25 10+ 4 cose
A lithotripter is used to break up a kidney stone by placing the kidney stone at one focus of an ellipsoid end-section and a source of shock waves at the focus of the other end-section. If the vertices of the end-sections are 30.0 cm apart and a minor axis of the ellipsoid is 6.0 cm, how far apart
A drain pipe 100 m long has an inside diameter d (in m) and an outside diameter D (in m). If the volume of material of the pipe itself is 0.50 m3, what is the equation relating d and D? Graph D as a function of d.
The stream from a fire hose follows a parabolic curve and reaches a maximum height of 60 ft at a horizontal distance of 95 ft from the nozzle. Find the equation that represents the stream, with the origin at the nozzle. Sketch the graph.
Show that the polar coordinate equation r = a sinθ + b cosθ represents a circle by changing it to a rectangular equation.
Use a calculator to view the circle x2 + y2 + 5y − 4 = 0.
To launch a spacecraft to the moon, it is first put into orbit around Earth and then into a parabolic path toward the moon. Assume the parabolic path is represented by x2 = 4 py and the spacecraft is later observed at (10, 3) (units in thousands of km) after launch. Will this path lead directly to
In the 1700s, the French physicist Reaumur established a temperature scale on which the freezing point of water was 0° and the boiling point was 80° . Set up an equation for the Fahrenheit temperature F (freezing point 32°, boiling point 212°) as a function of the Reaumur temperature R.
A missile is fired at an airplane and is always directed toward the airplane. The missile is traveling at twice the speed of the airplane. An equation that describes the distance r between the missile and the airplane iswhere θ is the angle between their directions at all times. See Fig. 21.119.
An ellipse has a focal property such that a light ray or sound wave emanating from one focus will be reflected through the other focus. Many buildings, such as Statuary Hall in the U.S. Capitol and the Taj Mahal, are built with elliptical ceilings with the property that a sound from one focus is
Two wheels in a friction drive assembly are equal ellipses, as shown in Fig. 21.90. They are always in contact, with the left wheel fixed in position and the right wheel able to move horizontally. Find the equation that can be used to represent the circumference of each wheel in the position shown.
Ohm’s law in electricity states that the product of the current i and the resistance R equals the voltage V across the resistance. If a battery of 6.00 V is placed across a variable resistor R, find the equation relating i and R and sketch the graph of i as a function of R.
For a constant capacitive reactance and a constant resistance, sketch the graph of the impedance and inductive reactance (as abscissas) for an alternating-current circuit.
A ray of light directed at one focus of a hyperbolic mirror is reflected toward the other focus. Find the equation that represents the hyperbolic mirror shown in Fig. 21.82.Fig. 21.82. F y 3.5 cm i F 2.8 cm X
In studying the photoelectric effect, an equation used for the rate R at which photoelectrons are ejected at various angles θ isSketch the graph. R = sin ²0 (10.5 cos0)²
What type of graph is represented by the equationsa.b.c. Are the equations in parts (a) and (b) functions? Explain. y = √√9 (x - 2)² ?
Find the relation between x and y such that (x, y) is always 3 units from the origin.
A road passes through a tunnel with a semielliptical cross section 64 ft wide and 18 ft high at the center. What is the height of the tallest vehicle that can pass through the tunnel at a point 22 ft from the center? See Fig. 21.67. Fig. 21.67 18 ft 22 ft 64 ft
The center of a regular hexagon is at the pole with one vertex at (2, π). What are the polar coordinates of the other vertices?
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.x² + y² = 94x² + y² = 16
The velocity of sound v increases 0.607 m/s for each increase in temperature T of 1.00°C. If v = 343m s for T = 20.0°C, express v as a function of T.
An agricultural test station is to be divided into rectangular sections,1 each with a perimeter of 480 m. Express the area A of each section in terms of its width w and identify the type of curve represented. Sketch the graph of A as a function of w. For what value of w is A the greatest?
Find the relation between x and y such that (x, y) is always equidistant from the y-axis and (2, 0).
Find the distance between the points (4, π/6) and (5, 5π/3).
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.y = exx² − y² = 1
What type of graph is represented by the equations(a) x2 + (y − 1)2 = 0? (b) x2 + (y − 1)2 = −1?
A wire is fastened 36.0 ft up on each of two telephone poles that are 200̄ ft apart. Halfway between the poles the wire is 30.0 ft above the ground. Assuming the wire is parabolic, find the height of the wire 50.0 ft from either pole.
Under certain conditions, the x- and y-components of a magnetic field B are given by the equationsWrite these equations in terms of polar coordinates. B₂ -ky x² + y² 2 and B y kx x² + y² 2
Noting the graphs in Exercises 17, 18, 37, and 38, what conclusion do you draw about the value of n and the graph of r = a sin nθ or r = a cos nθ?Data from Exercises 17r = 4sin 2θ (rose)
An acid solution is made from x liters of a 20% solution and y liters of a 30% solution. If the final solution contains 20 L of acid, find the equation relating x and y.
An architect designs a window in the shape of an ellipse 4.50 ft wide and 3.20 ft high. Find the perimeter of the window from the formula p = π(a + b). This formula gives a good approximation for the perimeter when a and b are nearly equal.
Maximum intensity for monochromatic (single-color) light from two sources occurs where the difference in distances from the sources is an integer number of wavelengths. Find the equation of the curves of maximum intensity in a thin film between the sources where the difference in paths is two
Show that the diagonals of a square are perpendicular to each other.
In designing a domed roof for a building, an architect uses the equationwhere k is a constant. Write this equation in polar form. x² + 2 2 k2 1,
A laser beam 6.80 mm in diameter is incident on a plane surface at an angle of 62.0°, as shown in Fig. 21.68. What is the elliptical area that the laser covers on the surface? (See Exercise 55.)Data from Exercises 55The ends of a horizontal tank 20.0 ft long are ellipses, which can be described by
Is the point (0.1, 3.1) inside, outside, or on the circle x2 + y2 − 2x − 4y + 3 = 0?
A wall is 15 cm thick. At the outside, the temperature is 3 C ° , and at the inside, it is 23°C. If the temperature changes at a constant rate through the wall, write an equation of the temperature T in the wall as a function of the distance x from the outside to a point inside the wall. What is
The ends of a horizontal tank 20.0 ft long are ellipses, which can be described by the equation 9x2 + 20y2 = 180, where x and y are measured in feet. The area of an ellipse is A = πab. Find the volume of the tank.
The center of a circle is (2,−3), and one end of a diameter is (−1, 2). What are the coordinates of the other end of the diameter?
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.x² − 4y² + 2x − 3 = 0y² − 4x − 4 = 0
Find the equation of the line along which the diameter of the circle x2 + y2 − 2x − 4y − 4 = 0 lies, if the diameter is parallel to the line 3x + 5y = 4.
An oil-storage tank is emptied at a constant rate. At 10 a.m., 1800 barrels remain, and at 2 p.m., 600 barrels remain. If pumping started at 8 a.m., find the equation relating the number of barrels n at time t (in h) from 8 a.m. When will the tank be empty?
The velocity v (in ft/s) of a jet of water flowing from an opening in the side of a certain container is given by v = 8√h, where h is the depth (in ft) of the opening. Sketch a graph of v vs. h.
A line segment has a slope of 3 and one endpoint at (−2, 5). If the other endpoint is on the x-axis, what are its coordinates?
The shape of a swimming pool can be described by the polar equation r = 5cosθ (dimensions in meters). Find the rectangular equation for the perimeter of the pool.
The polar equation of the path of a weather satellite of Earth iswhere r is measured in miles. Find the rectangular equation of the path of this satellite. The path is an ellipse, with Earth at one of the foci. r = 4800 1 + 0.14 cos 0
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.y = 2sin xy = 2 − x²
For the equation (x − h)2 + (y − k)2 = p, how does the value of p indicate whether the graph is a circle, a point, or does not exist? Explain.
In a research project on cancer, a tumor was determined to weigh 30 mg when first discovered. While being treated, it grew smaller by 2 mg each month. Find the equation relating the weight w of the tumor as a function of the time t in months. Graph the equation.
A small island is 4 km from a straight shoreline. A ship channel is equidistant between the island and the shoreline. Write an equation for the channel.
The points (−1, 3), (5, y), and (2, 4) are collinear (on the same line). Find y.
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.y = 4ln xxy = 6
Determine whether the graph of x2 + y2 − 2x + 10y + 29 = 0 is a circle, a point, or does not exist.
The length of a rectangular solar cell is 10 cm more than the width w. Express the perimeter p of the cell as a function of w. What is the meaning of the slope of the line?
Under certain circumstances, the maximum power P (in W) in an electric circuit varies as the square of the voltage of the source E0 and inversely as the internal resistance Ri (in Ω) of the source. If 10 W is the maximum power for a source of 2.0 V and internal resistance of 0.10Ω, sketch
Find the coordinates of the point on the y-axis that is equidistant from (−3,−5) and (2, 4).
The control tower of an airport is taken to be at the pole, and the polar axis is taken as due east in a polar coordinate graph. How far apart (in km) are planes, at the same altitude, if their positions on the graph are (6.10, 1.25) and (8.45, 3.74)?
View the curves of the given equations on a calculator.x2 + 3y + 2 − (1 + x)2 = 0
A 12-ft pole leaning against a wall slips to the ground. Find the equation that represents the path of the midpoint of the pole. See Fig. 21.40.Fig. 21.40. y 2y 0 (x, y) 2x -X
The inner and outer circles of the cross section of a pipe are represented by the equations 2.00x2 + 2.00y2 = 5.73 and 2.80x2 + 2.80y2 = 8.91. How thick (in in.) is the pipe wall?
A police report stated that a bullet caromed upward off a floor at an angle of 16.5° with the floor, as shown in Fig. 21.28. What is the equation of the bullet’s path after impact? 16.5° y Fig. 21.28 -X
Are the points (3, 5), (1,−1), and (6, 14) collinear?
The perimeter of a certain type of machine part can be described by the equation r = a sinθ + b cosθ(a > 0, b > 0). Explain why all such machine parts are circular.
A rose garden has two intersecting circular paths with a fountain in the center of the longer path. The paths can be represented by the equations x2 + y2 − 20x + 75 = 0 and x2 + y2 = 100 (all distances in meters) with the fountain at the origin. What are (a) The closest (b) The farthest
View the curves of the given equations on a calculator.y2 = 4x + 6
Are the points (−1, 6) and (5, 5) equidistant from (2, 1)?
View the curves of the given equations on a calculator.2x2 + 2y2 + 4y − 3 = 0
A triangular machine part has vertices at A(−3, 6), B(−1, 1), and C(5, 2). What is the length of the median from (−1, 1)? See Fig. 21.14. Α' -4 B y 4 4 Fig. 21.14 C X
In a hoisting device, two of the pulley wheels may be represented by x2 + y2 = 14.5 and x2 + y2 − 19.6y + 86.0 = 0. How far apart (in in.) are the wheels?
A survey of the traffic on a particular highway showed that the number of cars passing a particular point each minute varied linearly from 6:30 a.m. to 8:30 a.m. on workday mornings. The study showed that an average of 45 cars passed the point in 1 min at 7 a.m. and that 115 cars passed in 1 min at
View the curves of the given equations on a calculator.2x2 + (y − 3)2 − 5 = 0
After taking off, a plane gains altitude at 600 m/min for 5.0 min and then continues to gain altitude at 300 m/min for 15 min. It then continues at a constant altitude. Find the altitude h as a function of time t for the first 20 min, and sketch the graph of h = f(t).
Treat the given nonlinear functions as linear functions in order to sketch their graphs. At times, this can be useful in showing certain values of a function. For example, y = 2 + 3x2 can be shown as a straight line by graphing y as a function of x2 . A table of values for this graph is shown along
A wire is rotating in a circular path through a magnetic field to induce an electric current in the wire. The wire is rotating at 60.0 Hz with a constant velocity of 37.7 m/s. Taking the origin at the center of the circle of rotation, find the equation of the path of the wire.
View the curves of the given equations on a calculator.x2 + 2xy + y2 − 3x + 8y = 0
A communications satellite remains stationary at an altitude of 22,500 mi over a point on Earth’s equator. It therefore rotates once each day about Earth’s center. Its velocity is constant, but the horizontal and vertical components, vH and vV, of the velocity constantly change. Show that the
Denver is 1350 km east and 900 km south of Seattle. Edmonton is 620 km east and 640 km north of Seattle. How far is Denver from Edmonton?
View the curves of the given equations on a calculator.r = 3cos(3θ/2)
Treat the given nonlinear functions as linear functions in order to sketch their graphs. At times, this can be useful in showing certain values of a function. For example, y = 2 + 3x2 can be shown as a straight line by graphing y as a function of x2 . A table of values for this graph is shown along
Find the equation describing the rim of a circular porthole 2 ft in diameter if the top is 6 ft below the surface of the water. Take the origin at the water surface directly above the center of the porthole.
A person is working out on a treadmill inclined at 12% (the slope of the treadmill expressed in percent). What is the angle between the treadmill and the horizontal?
Treat the given nonlinear functions as linear functions in order to sketch their graphs. At times, this can be useful in showing certain values of a function. For example, y = 2 + 3x2 can be shown as a straight line by graphing y as a function of x2 . A table of values for this graph is shown along
View the curves of the given equations on a calculator.r = 5 − 2sin 4θ
An earthquake occurred 37° north of east of a seismic recording station. If the tremors travel at 4.8 km/s and were recorded 25 s later at the station, find the equation of the circle that represents the tremor recorded at the station. Take the station to be at the center of the coordinate system.
Treat the given nonlinear functions as linear functions in order to sketch their graphs. At times, this can be useful in showing certain values of a function. For example, y = 2 + 3x2 can be shown as a straight line by graphing y as a function of x2 . A table of values for this graph is shown along
View the curves of the given equations on a calculator.r = 2 − 3cscθ
In analyzing the strain on a beam, Mohr’s circle is often used. To form it, normal strain is plotted as the x-coordinate and shear strain is plotted as the y-coordinate. The center of the circle is midway between the minimum and maximum values of normal strain on the x-axis. Find the equation of
View the curves of the given equations on a calculator.r = 2sin(cos 3θ)
An architect designs a Norman window, which has the form of a semicircle surmounted on a rectangle, as in Fig. 21.42. Find the area (in m2) of the window if the circular part is on the circle x2 + y2 − 3.00y + 1.25 = 0.
Considering Eq. (21.30) of an ellipse, describe the graph if a = b.Data from Eq. (21.30)(major axis parallel to x -axis) (x - h)² a² + (y - k)² b2 = 1
Find the equation of the locus of a point P(x, y) that moves as stated.Always 4 units from (3,−4)
Find the equation of the locus of a point P(x, y) that moves as stated.Passes through (7,−5) with a constant slope of −2
Find the equation of the locus of a point P(x, y) that moves as stated.The sum of its distances from (1,−3) and (7,−3) is 8.
Find the equation of the locus of a point P(x, y) that moves as stated.The difference of its distances from (3,−1) and (3,−7) is 4.
Find the equation of the locus of a point P(x, y) that moves as stated.Its distance from y = 6 always equals its distance to (0,−6).
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