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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
In Example 4, change π/4 to π/6.Data from Example 4Using Eq. (21.41), we can transform the polar coordinates (4, π/4) into the rectangular coordinates (2√2,2√2), becauseSee Fig. 21.107. π x = 4 cos = 4 4 2 = 2√2 and T y = 4sin = 4(2)= = 2√2
In Example 3, change −9y2 to −y2 and then follow the same instructions as in Exercise 1.Data from Example 3Determine the coordinates of the vertices and foci of the hyperbola 4x2 − 9y2 = 36 First, by dividing through by 36, we haveFrom this form, we see that a2 = 9 and b2 = 4. In turn, this
In Example 2, change (2, 1) to (−2, 1).Data from Example 2Find the equation of the circle with center at (2, 1) and that passes through (4,−6). In Eq. (21.11), we can determine the equation of this circle if we can find h, k, and r. From the given information, the center is (2, 1), which means
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 1²+ y² = 1 4
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. 25 正 = 1 144
In Example 4, change the + before 4x to −.Data from Example 4Find the slope and the y-intercept of the straight line for which the equation is 2y + 4x − 5 = 0. We write this equation in slope-intercept form:Because the coefficient of x in this form is −2, the slope is −2. The constant on
In Example 4, change cos to sin.Data from Example 4Plot the graph of r = 2cos2θ.In finding values of r, we must be careful first to multiply the values of θ by 2 before finding the cosine of the angle. Also, for this reason, we take values of θ as multiples of π/12, so as to get enough useful
In Example 5(b), change −1.732 to −0.5774.Data from Example 5(b)If a line has a slope of −1.732, we know that tan α = −1.732. Because tan α is negative, α must be a second-quadrant angle. Therefore, using a calculator to show that tan60° = 1.732, we find that α = 180° − 60° =
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.x2 + 2y2 − 2 = 0
Sketch the graph of the straight line 4x − 2y + 5 = 0 by finding its slope and y-intercept.
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. x² y = 1 16 4
In Example 5, change 3 to −3 and −5 to 5.Data from Example 5Using Eq. (21.42), we can transform the rectangular coordinates (3,−5) into polar coordinates, as follows.We know that θ is a fourth-quadrant angle because x is positive and y is negative. Therefore, the point (3,−5) in
Transform the given equations by rotating the axes through the given angle. Identify and sketch each curve.8x2 − 4xy + 5y2 = 36, θ = tan−1 2
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 2 X 100 64 = 1
In Example 5, change the + before 2y to −.Data from Example 5Find the general form of the equation of the line parallel to the line 3x + 2y − 6 = 0 and that passes through the point (−1, 2). Because the line whose equation we want is parallel to the line 3x + 2y − 6 = 0, it has the same
Determine each of the following as being either true or false. If it is false, explain why.The center of the circle x2 + y2 + 2x + 4y + 5 = 0 is (1, 2).
In Example 7(a), change B(6, 0) to B(−4,−2).Data from Example 7(a)Show that the line segments joining A(−5, 3), B(6, 0), and C(5, 5) form a right triangle. See Fig. 21.12.If these points are vertices of a right triangle, the slopes of two of the sides must be negative reciprocals. This would
In Example 3(b), change −9y to 7y.Data from Example 3(b)The parabola 2x2 = −9y fits the form of Eq. (21.16) if we write it in the formHere, we see that 4 p = −9/2. Therefore, its axis is along the y-axis, and its vertex is at the origin. Because 4 p = −9/2, we haveThe parabola opens
In Example 7, what is the decibel loss when the sound approaches the microphone 120° from the point directly in front of the microphone?Data from Example 7A cardioid microphone, commonly used for vocals or speech, is designed to pick up sound in front of the microphone but reject sound coming from
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve. (x + 4)² 4 + (y − 1)² = 1 -
In Example 7, change the + before 6y to −.Data from Example 7Display the graph of the circle 3x 2 + 3y 2 + 6y − 20 = 0 on a calculator. To fit the form of a quadratic equation in y, we write 3y2 + 6y + (3x2 − 20) = 0. Now, using the quadratic formula to solve for y, we letwhich means we get
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 외 25 144 1
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. X 공무 9 1 = 1
Transform the given equations by rotating the axes through the given angle. Identify and sketch each curve.2x2 + 24xy − 5y2 = 8, θ = tan−1 3/4
Determine each of the following as being either true or false. If it is false, explain why.The directrix of the parabola x2 = 8y is the line y = 2.
Determine the center and the radius of each circle.(x − 2)2 + (y − 1)2 = 25
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve. (x - 1)² 4 (y-2)² 25
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.2x2 − y2 − 1 = 0
Find the vertex and the focus of the parabola x2 = −12y. Sketch the graph.
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (−3, 8) with a slope of 4.
Find the distance between the given pairs of points.(3, 8) and (−1,−2)
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 49 y² 81 1
Identify the type of curve that each equation represents by evaluating B2 − 4AC.x2 + 2xy + x − y − 3 = 0
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. y² X = 1 4 21
Determine each of the following as being either true or false. If it is false, explain why.The vertices of the ellipse 9x2 + 4y2 = 36 are (2,0) and (−2, 0).
Plot the given polar coordinate points on polar coordinate paper.(3, π/6)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.y2 = 4x
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.y(y + x2) = 4(x + y2)
Find the equation of the circle with center at (−1, 2) and that passes through (2, 3).
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (−2,−1) with a slope of −2.
Find the distance between the given pairs of points.(−1, 3) and (−8,−4)
Identify the type of curve that each equation represents by evaluating B2 − 4AC.8x2 − 4xy + 2y2 + 7 = 0
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 4x² 25 + 9y² 4 = 1
Plot the given polar coordinate points on polar coordinate paper.(2, π)
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. 4x² y² 25 4
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve.(y + 5)2 = −8(x − 2)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.y2 = 16x
Plot the curves of the given polar equations in polar coordinates.r = −2
Plot the given polar coordinate points on polar coordinate paper. 5 2π 5 la
Determine the center and the radius of each circle.4(x + 1)2 + 4y2 = 121
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.8x2 + 2y2 = 6y(1 − y)
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve. (x + 1)² + y² 36 = 1
Find the equation of the straight line that passes through (−4, 1) and (2,−2).
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (2,−5) and (4, 2).
Find the distance between the given pairs of points.(4,−5) and (4,−8)
Identify the type of curve that each equation represents by evaluating B2 − 4AC.x2 − 2xy + y2 + 3y = 0
Determine each of the following as being either true or false. If it is false, explain why.The equation x2 = (y − 1)2 represents a hyperbola.
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve. 9y² - x² = 1 25
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.y2 = −64x
Plot the curves of the given polar equations in polar coordinates.θ = 3π/4
Determine the center and the radius of each circle.9x2 + 9(y − 6)2 = 64
Where is the focus of a parabolic reflector that is 12.0 cm across and 4.00 cm deep?
Determine each of the following as being either true or false. If it is false, explain why.The equation 5x2 − 8xy + 5y2 = 9 represents an ellipse.
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve. (y-4)²(x + 2)² 49 4 = 1
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Has an x-intercept (4, 0) and a y-intercept of (0,−6).
Find the distance between the given pairs of points.(−3,−7) and (2, 10)
Identify the type of curve that each equation represents by evaluating B2 − 4AC.4xy + 3y2 − 8x + 16y + 19 = 0
Plot the given polar coordinate points on polar coordinate paper.(5, −π/3)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.y2 = −36x
Plot the curves of the given polar equations in polar coordinates.θ = −1.5
Find the equation of each of the circles from the given information.Center at (0, 0), radius 3
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.4x2 + 9y2 = 324
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.9x2 − y2 = 4
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.2.2x2 − (x + y) = 1.6
Find the equation of each of the circles from the given information.Center at (−3, 5) and tangent to line y = 10
A hallway 16 ft wide has a ceiling whose cross section is a semiellipse. The ceiling is 10 ft high at the walls and 14 ft high at the center. Find the height of the ceiling 4 ft from each wall.
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (−7, 12) with an inclination of 45°.
Find the distance between the given pairs of points.(−12, 20) and (32,−13)
Identify the type of curve that each equation represents by evaluating B2 − 4AC.13x2 + 10xy + 13y2 + 6x − 42y − 27 = 0
Determine each of the following as being either true or false. If it is false, explain why.The rectangular equation x = 2 represents the same curve as the polar equation r = 2secθ.
Plot the given polar coordinate points on polar coordinate paper.(−8, 7π/6)
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve.(x + 3)2 = −12(y − 1)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.x2 = 72y
Plot the curves of the given polar equations in polar coordinates.r = 4secθ
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve. x2 0.16 + (y + 1)² 0.25 = 1
Find the equation of each of the circles from the given information.Center at (0, 0), radius 12
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.9x2 + 36y2 = 144
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.4x2 − 9y2 = 6
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.2x2 + 4y2 = y + 2x
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Has a y-intercept (0,−2) and an inclination of 120°.
Find the distance between the given pairs of points.(23,−9) and (−25, 11)
Identify the type of curve that each equation represents by evaluating B2 − 4AC.x2 − 4xy + 4y2 + 36x + 28y + 24 = 0
Determine each of the following as being either true or false. If it is false, explain why.The graph of the polar equation θ = π/4 is a straight line.
Plot the given polar coordinate points on polar coordinate paper.(−5, π/4)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.x2 = y
Find the equation of each of the circles from the given information.Center at (2, 3), radius 4
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.49x2 + 8y2 = 196
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.x2 = (y − 1)(y + 1)
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.2y2 − 5x2 = 20
Find the center and vertices of the conic section 4y2 − x2 − 4x − 8y − 4 = 0. Show completely the sketch of the curve.
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