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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Explain how a branch of a hyperbola differs from a parabola.
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r = 4cosθ + 2sinθ
What is the general form of the equation of a family of parabolas if each vertex and focus is on the x-axis?
Find the area of the triangle in Exercise 42.Data from Exercises 42(−1, 3), (3, 5), and (5, 1) are the vertices of a right triangle.
Using a calculator, show that the curves r = 2sinθ and r = 2cosθ intersect at right angles. Proper window settings are necessary.
Find the polar equation of each of the given rectangular equation.y = 2x
Find the area bounded between the two circles given by x2 + y2 = 9 and x2 + y2 = 25.
Find the perimeter of the triangle in Exercise 41.Data from Exercises 41(2, 3), (4, 9), and (−2, 7) are vertices of an isosceles triangle.
Solve the given problem. All coordinates given are polar coordinates.Is the point (2, 3π/4) on the curve r = 2sin2θ?
View the curves of the given polar equations on a calculator.r csc 5θ = 3
Plot the given curves in polar coordinates.r = 1 − 3cosθ
Determine the value of k.The distance between (k, 0) and (0, 2k) is 10.
For what values of k does the ellipse x2 + ky2 = 1 have its vertices on the y-axis? Explain how these values are found.
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.5x2 + 12y + 18 = 2y2
Find the polar equation of each of the given rectangular equations.xy = 9
Use a calculator to view the hyperbola x2 − 4y2 + 4x + 32y − 64 = 0.
The equation of a parabola with vertex (h, k) and axis parallel to the y-axis is (x − h)2 = 4 p(y − k). Sketch the parabola for which (h, k) is (−1, 2) and p = −3.
Determine whether the circles with the given equations are symmetric to either axis or the origin.x2 + y2 − 4x − 5 = 0
Find k if the lines given in Exercise 37 are perpendicular.Data from Exercises 37Find k if the lines 4x − ky = 6 and 6x + 3y + 2 = 0 are parallel.
Use the given values to determine the type of curve represented.In Eq. (21.34), if A > C > 0 and B = D = E = F = 0, describe the locus of the equation.Eq. 21.34.Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
For what value of k does the ellipse x 2 + k2y2 = 25 have a focus at (3, 0)? Explain how this value is found.
Use a calculator to view the hyperbola 5y2 − 4x2 + 8x + 40y + 56 = 0.
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r = sinθ
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.9x2 + 9y2 + 14 = 6x + 24y
Determine the value of k.Points (6,−1), (3, k), and (−3,−7) are on the same line.
Plot the given curves in polar coordinates.r = 4 cos 3θ
View the curves of the given polar equations on a calculator.r = 2cosθ + 3sinθ
The chord of a parabola that passes through the focus and is parallel to the directrix is called the latus rectum of the parabola. Find the length of the latus rectum of the parabola y 2 = 4 px.
Determine whether the circles with the given equations are symmetric to either axis or the origin.3x2 + 3y2 + 24y = 8
Find k if the lines 3x − y = 9 and kx + 3y = 5 are perpendicular. Explain how this value is found.
Show that the ellipse 2x2 + 3y2 − 8x − 4 = 0 is symmetric to the x-axis.
The equation of a hyperbola with center (h, k) and transverse axis parallel to the x-axis isSketch the hyperbola that has a transverse axis of 4, a conjugate axis of 6, and for which (h, k) is (−3, 2). (x - h)² a² (y - k)² 79 = 1.
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r secθ = 4
In Example 10, change the + in the middle member to − and then display the solution on a graphing calculator. Data from Example 10 Display the solution of the inequality −1 Then enter y1 = −1 NORMAL FLOAT AUTO REAL DEGREE MP Plot1 Plot2 Plot3 NY1E-1
In Example 9(b), change the −1 to −3 and the 3 to 1 and then write the two forms in which an inequality represents the statement.Data from Example 9(b)To state that a number x may be greater than −1 and also less than or equal to 3, we write −1 < x ≤ 3. (It can also be written as x
Draw a sketch of the graph of the given inequality.2y < 3x − 2
Find the indicated maximum and minimum values by the linear programming method of this section, the constraints are shown below the objective function.Maximum P: P = 5x + 2y x ≥ 0, y ≥ 0 2x + y ≤ 6
Graphing the constraints of a linear programming problem shows the consecutive vertices of the region of feasible points to be (1, 3), (8, 0), (9, 7), (5, 8), (0, 6), and (1, 3). What are the maximum and minimum values of the objective function F = 2x + 5y in this region?
Solve the given inequalities. Graph each solution.|x + 4| < 6
Determine each of the following as being either true or false. If it is false, explain why.The solution of the inequality |x − 2| < 5 is x < −3 or x > 7.
Solve the given inequalities algebraically and graph each solution.− 1 < 1 − 2x < 5
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.x2 + 3x ≥ 0
Draw a sketch of the graph of the given inequality.y ≥ 2x + 5
Solve the given inequalities. Graph each solution.x − 3 > −4
Solve the given inequalities algebraically and graph each solution. x² + x x-2 ≤0
For the inequality 4 < 9, state the inequality that results when the given operations are performed on both members.Add 5.
Solve the given inequalities. Graph each solution.|5x + 4| > 6
Find the indicated maximum and minimum values by the linear programming method of this section, the constraints are shown below the objective function.Maximum P: P = 2x + 7y x ≥ 1, y ≥ 0 x + 4y ≤ 8
Determine each of the following as being either true or false. If it is false, explain why.The graphical solution of the inequality y ≥ x + 1 is shown as all points above the line y = x + 1.
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.2x2 ≥ 4x
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.x2 − 4x < 21
Solve the given inequalities. Graph each solution. | 12N-1 > 3
Draw a sketch of the graph of the given inequality.y ≤ 15 − 3x
Solve the given inequalities. Graph each solution.3x + 2 ≤ 11
For the inequality 4 < 9, state the inequality that results when the given operations are performed on both members.Subtract 16.
Find the indicated maximum and minimum values by the linear programming method of this section, the constraints are shown below the objective function.Maximum P: P = 5x + 9y x ≥ 0, y ≥ 0 x + 2y < 6
Determine each of the following as being either true or false. If it is false, explain why.The maximum value of the objective function F = x + 2y, subject to the conditions x ≥ 0, y ≥ 0, 2x + 3y ≤ 6, is 4.
Solve the given inequalities algebraically and graph each solution.|2x + 1| ≥ 3
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.2x2 − 12 ≤ −5x
Draw a sketch of the graph of the given inequality.3x + 2y + 6 > 0
Solve the given inequalities. Graph each solution.1/2x < 32
For the inequality 4 < 9, state the inequality that results when the given operations are performed on both members.Multiply by 4.
Find the indicated maximum and minimum values by the linear programming method of this section, the constraints are shown below the objective function.Minimum C: C = 10x + 20y x ≥ 0, y ≥ 0 3x + 5y > 30
Solve the given inequalities. Graph each solution.1 + |6x − 5| ≤ 5
Solve each of the given inequalities algebraically. Graph each solution.2x − 12 > 0
Solve the given inequalities algebraically and graph each solution.|2 − 3x| < 8
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.9t2 + 6t > −1
Draw a sketch of the graph of the given inequality.x + 4y − 8 < 0
Solve the given inequalities. Graph each solution.−4t > 12
For the inequality 4 < 9, state the inequality that results when the given operations are performed on both members.Multiply by −2.
Find the indicated maximum and minimum values by the linear programming method of this section, the constraints are shown below the objective function.Minimum C: C = 5x + 6y x ≥ 0, y ≥ 0 x + y ≥ 5 x + 2y > 7
Solve the given inequalities. Graph each solution.|30 − 42x| ≤ 0
Solve each of the given inequalities algebraically. Graph each solution.2.4(T − 4.0) ≥ 5.5 − 2.4T
Sketch the region in which the points satisfy the following system of inequalities:y < x2y ≥ x + 1
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.x2 + 4x ≤ −4
Draw a sketch of the graph of the given inequality.4y < x2
Solve the given inequalities. Graph each solution.3x − 5 ≤ −11
Determine the values of x for which represents a real number. 9-x- zx^
For the inequality 4 < 9, state the inequality that results when the given operations are performed on both members.Divide by −1.
Solve the given inequalities. Graph each solution.|3 − 4x| > 3
Solve each of the given inequalities algebraically. Graph each solution.4 < 2x − 1 < 11
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.6x2 + 1 < 5x
In Example 2, change the − sign in the numerator to +.Data from Example 2Simplifyby expressing the result in terms of one-half the given angle. Then, using a calculator, show that the values are equal. We note that the given expression fits the form of the right side of Eq. (20.26), which means
Solve the given inequalities. Graph each solution.32 − 5x < −8
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.3x3 − 6x2 + 3x ≥ 0
In Example 2, change 5/13 to 12/13 and then find the value of cos(α + β).Data from Example 2Given that sinα = 5/13 (α in the first quadrant) and sin β = − 3/5 (for β in the third quadrant), find cos(α + β).Because sin α = 5/13 for α in the first quadrant, from Fig. 20.7, we have cos α
Solve the given quadratic inequalities. Check each by displaying the solution on a calculator.|x2 + x − 4| > 2[After using Eq. (17.1), you will have two inequalities. The solution includes the values of x that satisfy either of the inequalities.]Data from Eq. (17.1)If |f(x)| > n, then f (x)
In Example 1, change 2cosθ to tanθ.Data from Example 1Solve the equation 2 cosθ − 1 = 0 for all values of θ such that 0 ≤ θ ≤ 2π. Solving the equation for cosθ, we obtain cosθ = 1/2. The problem asks for all values of θ from 0 to 2π that satisfy the equation. We know that the
Solve the given quadratic inequalities. Check each by displaying the solution on a calculator.|x2 + x − 4| < 2[Use Eq. (17.2), and then treat the resulting inequality as two inequalities of the form f (x) > −n and f (x) < n. The solution includes the values of x that satisfy both of
In Example 1(d), change π/6 to π/3 and then evaluate tan 2π/3.Data from Example 1(d)If α = π/6 we have tan = 12 (7) tan 2 = 2 tan 1 - tan² (4) = 2(√3/3) 1 - (√3/3)² = √3 using Eq. (20.25)
Expand and simplify the given expressions by use of the binomial formula.(2x − 3)4
In Example 6, change 180° + x to 180° − x and then determine what other changes result.Data from Example 6Prove that sin(180° + x) = −sin x.Although x may or may not be an acute angle, this agrees with the results for the sine of a third-quadrant angle as discussed in Section 8.2. See Fig.
In Example 4, change the right side to tan x/sec x .Data from Example 4In proving the identitywe know that cot x = cos x/ sin x. Because sin x appears on the left, substituting for cot x on the right will eliminate cot x and introduce sin x. This should help us proceed in proving the identity.
In Example 2, change 2cos2 x to 2sin2 x.Data from Example 2Solve the equation 2 cos2 x − sin x − 1 = 0 (0 ≤ x < 2π).By use of the identity sin2 x + cos2 x = 1, this equation may be put in terms of sin x only. Thus, we haveSetting each factor equal to zero, we find sin x = 1 2, or sin x =
In Example 5, change 8/15(180° < α < 270°) to − 8/15(270° < α < 360°).Data from Example 5Given that tan α = 8/15 (180°< α < 270°), find cos(α/2).Knowing that tan α = 8/15 for a third-quadrant angle, we determine from Fig. 20.20 that cos α= −15/17. This
In Example 1(b), change 2x to 3A.Data from Example 1(b)y = tan−1 2x is read as “y is the angle whose tangent is 2x.” In this case, 2x = tan y.
Determine each of the following as being either true or false. If it is false, explain why.tan 2θ = cos2θ/sin2θ
In Example 3(a), change 1/2 to −1.Data from Example 3(a)This is the only value of the function that lies within the defined range. The value 5π/6 is not correct, even though sin(5π/6) = 1/2. Since 5π/6 lies outside the defined range. sin ||
In Example 6, change the first term on the left to sin y/cos3 y .Data from Example 6Prove the identityBecause the more complicated side is on the left, we will change the left side to tan y, the form on the right. Because we want tan y as the final form, we will use the basic identities to
Solve for x (0 ≤ x < 2π) analytically, using trigonometric relations where necessary: sin 2x + sin x = 0.
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