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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Find the equation of the locus of a point P(x, y) that moves as stated.A standard form conic that passes through (−3, 0) and (0, 4).
Show that the ellipse x2 + 9y2 = 9 has the same foci as the hyperbola x2 − y2 = 4.
The points (−2,−5), (3,−3), and (13, x) are collinear. Find x.
For the ellipse in Fig. 21.120, show that the product of the slopes PA and PB is −b2/a2. Fig. 21.120 B(-a, 0) y (0, b) (0, -b) P(x, y) A(a,0) X
For the polar coordinate point (−5, π/4), find another set of polar coordinates such that r < 0 and −2π < θ < 0.
Find the distance between the polar coordinate points (3, π/6) and (6,− π/3).
Show that the parametric equations y = cotθ and x = cscθ define a hyperbola.
In two ways, show that the line segments joining (−3, 11), (2,−1), and (14, 4) form a right triangle.
Find the equation of the circle that passes through (3,−2), (−1,−4), and (2,−5).
What type of curve is represented by (x + jy) 2 + (x − jy) 2 = 2? ( j = √−1)
Find the area of the square that can be inscribed in the ellipse 7x2 + 2y2 = 18.
An elliptical tabletop is 4.0 m long and has a 3.0 m by 2.0 m rectangular design inscribed in it lengthwise. See Fig. 21.121. What is the area of the tabletop? (The area of an ellipse is A = πab.) 2.0 m 3.0 m -4.0 m- Fig. 21.121
Using a graphing calculator, determine the number of points of intersection of the polar curves r = 4|cos 2θ| and r = 6sin[cos(cos3θ)].
By means of the definition of a parabola, find the equation of the parabola with focus at (3, 1) and directrix the line y = −3. Find the same equation by the method of translation of axes.
For what value(s) of k does x2 − ky2 = 1 represent an ellipse with vertices on the y-axis?
The total resistance RT of two resistances in series in an electric circuit is the sum of the resistances. If a variable resistor R is in series with a 2.5-Ω resistor, express RT as a function of R and sketch the graph.
A laser source is 2.00 cm from a spherical surface of radius 3.00 cm, and the laser beam is tangent to the surface. By placing the center of the sphere atthe origin, and the source on the positive x-axis, find the equation of the line along which the beam shown in Fig. 21.123 is directed.
An elliptical hot tub is twice as long as it is wide. If its length is 3.6 m, find the distance across the shorter span of the hot tub 1.0 m from the center. See Fig. 21.122. 3.6 m Fig. 21.122 -1.0 m
The acceleration of an object is defined as the change in velocity v divided by the corresponding change in time t. Find the equation relating the velocity v and time t for an object for which the acceleration is 20 ft/s2 and v = 5.0 ft/s when t = 0s.
The velocity v of a crate sliding down a ramp is given by v = v0 + at, where v0 is the initial velocity, a is the acceleration, and t is the time. If v0 = 5.75ft/s and v = 18.5ft/s when t = 5.50s, find v as a function of t. Sketch the graph.
An airplane touches down when landing at 100 mi/h. Its velocity v while coming to a stop is given by v = 100 − 20,000t, where t is the time in hours. Sketch the graph of v vs. t.
It takes 2.010 kJ of heat to raise the temperature of 1.000 kg of steam by 1.000°C. In a steam generator, a total of y kJ is used to raise the temperature of 50.00 kg of steam from 100°C to T°C. Express y as a function of T and sketch the graph.
The temperature in a certain region is 27°C, and at an altitude of 2500 m above the region it is 12°C. If the equation relating the temperature T and the altitude h is linear, find the equation.
The radar gun on a police helicopter 490 ft above a multilane highway is directed vertically down onto the highway. If the radar gun signal is cone-shaped with a vertex angle of 14°, what area of the highway is covered by the signal?
A circular wind turbine with a diameter of 90 m is attached to the top of a 110-m pole. Find the equation of the circle traced by the tips of the blades if the origin is at the bottom of the pole.
The quality factor Q of a series resonant electric circuit with resistance R, inductance L, and capacitance C is given bySketch the graph of Q and L for a circuit in which R = 1000Ω and C = 4.00μF. 7 I 2 = 1/2 √²/2 RVC
The arch of a small bridge across a stream is parabolic. If, at water level, the span of the arch is 80 ft and the maximum height above water level is 20 ft, what is the equation that represents the arch? Choose the most convenient point for the origin of the coordinate system.
A motorcycle cost $12,000 when new and then depreciated linearly $1250/year for four years. It then further depreciated linearly $1000/year until it had no resale value. Write the equation for the motorcycle’s value V as a function of t and sketch the graph of V = f(t).
The temperature of ocean water does not change with depth very much for about 300 m, and then as depth increases to about 1000 m, it decreases rapidly. Below 1000 m the temperature decreases very slowly with depth. A typical middle latitude approximation would be T = 22°C for the first 300 m of
The top horizontal cross section of a dam is parabolic. The open area within this cross section is 80 ft across and 50 ft from front to back. Find the equation of the edge of the open area with the vertex at the origin of the coordinate system and the axis along the x-axis.
A study indicated that the fraction f of cells destroyed by various dosages d of X-rays is given by the graph in Fig. 21.124. Assuming that the curve is a quarter-ellipse, find the equation relating f and d for 0 ≤ f ≤ 1 and 0 < d ≤ 10 units. Fig. 21.124 0.5 0 f 5 Ld 10
At very low temperatures, certain metals have an electric resistance of zero. This phenomenon is called superconductivity. A magnetic field also affects the superconductivity. A certain level of magnetic field HT, the threshold field, is related to the thermodynamic temperature T by HT/H0 = 1
A rectangular parking lot is to have a perimeter of 600 m. Express the area A in terms of the width w and sketch the graph.
The electric power P (in W) supplied by a battery is given by P = 12.0i − 0.500i2 , where i is the current (in A). Sketch the graph of P vs. i.
A 60-ft rope passes over a pulley 10 ft above the ground, and a crate on the ground is attached at one end. The other end of the rope is held at a level of 4 ft above the ground and is drawn away from the pulley. Express the height of the crate over the ground in terms of the distance the person is
The Colosseum in Rome is in the shape of an ellipse 188 m long and 156 m wide. Find the area of the Colosseum. (A = πab for an ellipse.)
A specialty electronics company makes an ultrasonic device to repel animals. It emits a 20–25 kHz sound (above those heard by people), which is unpleasant to animals. The sound covers an elliptical area starting at the device, with the longest dimension extending 120 ft from the device and the
A machine-part designer wishes to make a model for an elliptical cam by placing two pins in a design board, putting a loop of string over the pins, and marking off the outline by keeping the string taut. (Note that the definition of the ellipse is being used.) If the cam is to measure 10 cm by 6
Soon after reaching the vicinity of the moon, Apollo 11 (the first spacecraft to land a man on the moon) went into an elliptical lunar orbit. The closest the craft was to the moon in this orbit was 70 mi, and the farthest it was from the moon was 190 mi. What was the equation of the path if the
The vertical cross sections of two pipes as drawn on a drawing board are shown in Fig. 21.127. Find the polar equation of each. Fig. 21.127 2.40 cm y 3.80 cm
An electronic instrument located at point P records the sound of a rifle shot and the impact of the bullet striking the target at the same instant. Show that P lies on a branch of a hyperbola.
Tremors from an earthquake are recorded at the California Institute of Technology (Pasadena, California) 36 s before they are recorded at Stanford University (Palo Alto, California). If the seismographs are 510 km apart and the shock waves from the tremors travel at 5.0 km/s, what is the curve on
The path of a certain plane is r = 200(secθ + tanθ)−5 cosθ, 0 < θ < π/2 Sketch the path and check it on a calculator.
Under a force that varies inversely as the square of the distance from an attracting object (such as the sun exerts on Earth), it can be shown that the equation of the path an object follows is given in general by 1/r = a + b cos θ where a and b are constants for a particular path. First,
The sound produced by a jet engine was measured at a distance of 100 m in all directions. The loudness of the sound d (in decibels) was found to be d = 115 + 10 cosθ, where the 0° line for the angle θ is directed in front of the engine. Sketch the graph of d vs. θ in polar coordinates (use d as
Simplify the given expressions. + cos 6x 2
Evaluate exactly the given expressions if possible.cos−1 2
Factor and simplify.sinθ − sinθ cos2θ
Factor and simplify.sin3 t cos t + sin t cos3 t
Simplify the given expressions.Sin(90° − x)
Evaluate the indicated functions with the given information.Find sin 2x if cos x = 4/5 (in first quadrant).
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.2 cos2 x − 2 cos 2x − 1 = 0
Evaluate exactly the given expressions if possible.tan−1(− √3)
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.sin12°cos38° + cos12°sin38°
Factor and simplify.tan2 usec2 u − tan4 u
Simplify the given expressions.cos(3/2π − x)
Find cos 2x if sin x = −12/13 (in third quadrant).
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.csc2 x + 2 = 3 csc x
Simplify the given expressions. 4 - 4 cos 100
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.cos2 148° − sin2 148°
Evaluate exactly the given expressions if possible.sec−1(−2)
Factor and simplify.csc4y − 1
Evaluate the indicated functions with the given information.Find tan 2x if sin x = 0.5 (in second quadrant).
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.4 tan x − sec2 x = 0
Simplify the given expressions. 18+18 cos 1.4x
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.2sin π/14 cos π/14
Evaluate exactly the given expressions if possible.sec−1 0.5
Factor and simplify.sin x + sin x cot2x
Simplify the given expressions.sin (x + π/2)
Prove the given identities. sinx tan x = cos.x
Evaluate the indicated functions with the given information.Find sin 4x if tan x = −0.6 (in fourth quadrant).
Simplify the given expressions. 2sin²+ cos.x
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.sin(x − π/4) = cos(x − π/4)
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.1 − 2sin2 π/8
Evaluate exactly the given expressions if possible.cot−1√3
Simplify the given expressions.sin3x cos(3x − π) − cos3x sin(3x − π)
Prove the given identities. csc 0 sec = cote
Simplify the given expressions.6 sin 5x cos 5x
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.sin 2x cos x − cos 2x sin x = 0
Simplify the given expressions. 2 cos²sec 0
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.cos73°cos(−142°) + sin 73°sin(−142°)
Evaluate exactly the given expressions if possible.sin−1(−√2/2)
Simplify the given expressions.cos(x + π)cos(x − π) + sin(x + π)sin(x − π)
Simplify the given expressions.4sin2 x cos2 x
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.cos3x cos x − sin 3x sin x = 0
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the simplified expression.cos3°cos215° − sin3°sin 215°
Evaluate exactly the given expressions if possible.cos−1(−√3/2)
Prove the given identities.sin x sec x = tan x
Evaluate each expression by first changing the form. Verify each by use of a calculator.sin122°cos32° − cos122°sin32°
Find the value of sin(α/2) if cos α = 12/13 (0° < α < 90°).
Simplify the given expressions.1 − 2sin2 4x
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.tan x + 1 = 0
Prove the given identities. sin 20 1 + cos20 = tan 0
Evaluate exactly the given expressions if possible.sin(tan−1√3)
Derive the given equations by letting α + β = x and α − β = y, which leads to α = 1/2(x + y) and β = 1/2(x − y). The resulting equations are known as the factor formulas.Use Eq. (20.16) and the substitutions above to derive the equationData from Eq. (20.16) cos x cos y = -2 sin(x + y)
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result.cos7x cos3x + sin 7x sin3x
Prove the given identities. 0 0 sin-cos- 2 2 sin 0 2
Prove the given identities.tan x + cot x = sec x csc x
Prove the given identities.cos2 α − sin2 α = 1 − 2sin2 α
Solve the system of equations r = sinθ, r = sin 2θ, for 0 ≤ θ < 2π.
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