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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

(a) Use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function to the nearest thousandth. Use the zero or root
Use long division to divide. 1. (30x2 − 3x + 8) / (5x - 3) 2. (4x + 7) / (3x - 2) 3. (5x3 - 21x2 - 25x - 4) / (x2 - 5x - 1)
Use synthetic division to divide. 1. (2x3 − 25x2 + 66x + 48) / (x - 8) 2. (5x3 + 33x2 + 50x - 8) / (x + 4) 3. (x4 - 2x2 + 9x) / (x + 3) 4. 6x4 - 4x3 - 27x2 + 18x) / (x - 2)
Use the Remainder Theorem and synthetic division to find each function value. f (x) = x4 + 10x3 - 24x2 + 20x + 44 (a) f (-3), (b) f (-1)
Use synthetic division to determine whether the given values of x are zeros of the function. f (x) = 20x4 + 9x3 − 14x2 − 3x (a) x = −1 (b) x = 34 (c) x = 0 (d) x = 1
(a) verify the given factor(s) of f (x), (b) find the remaining factors of f (x), (c)use your results to write the complete factorization of f (x), (d)list all real zeros of f, and (e) confirm your results by using a graphing utility to graph the
Determine the number of zeros of the polynomial function. 1. f (x) = x - 6 2. g (x) = x2 − 2x − 8 3. h (t) = t2 − t5 4. f (x) = x8 + x9 5. f (x) = (x − 8)3 6. g (t) = (2t − 1)2 − t4
Find the rational zeros of the function. 1. f (x) = x3 + 3x2 − 28x − 60 2. f (x) = x3 − 10x2 + 17x − 8 3. f (x) = 3x3 + 8x2 − 4x - 16
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 1. 2/3, 4, √3i 2. 2, -3, 1 - 2i
Use the given zero to find all the zeros of the function. Function ........................................................................... Zero 1. h (x) = −x3 + 2x2 − 16x + 32 ................................................. −4i 2. g (x) = 2x4 − 3x3 − 13x2 + 37x - 15
Write the polynomial as the product of linear factors and list all the zeros of the function. 1. f (x) = x3 + 4x2 − 5x 2. g (x) = x3 − 7x2 + 36 3. g (x) = x4 + 4x3 − 3x2 + 40x + 208 4. f (x) = x4 + 8x3 + 8x2 − 72x - 153
Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. 1. f (x) = x3 − 16x2 + x − 16 2. f (x) = 4x4 −
Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. 1. g (x) = 5x3 + 3x2 − 6x + 9 2. h (x) = −2x5 + 4x3 − 2x+ + 5
Use synthetic division to verify the upper and lower bounds of the real zeros of f. 1. f (x) = 4x3 − 3x2 + 4x - 3 (a) Upper: x = 1 (b) Lower: x = −1/4 2. f (x) = 2x3 − 5x2 − 14x + 8 (a) Upper: x = 8 (b) Lower: x = −4
A right cylindrical water bottle has a volume of 36 cubic inches and a height nine inches greater than its radius. Find the dimensions of the water bottle.
A kitchen has a volume of 60 cubic meters. The width of the room is one meter greater than the length and the height is one meter less than the length. Find the dimensions of the room.
(a) Find the zeros of each quadratic function g(x). (i) g(x) = x2 − 4x − 12 (ii) g(x) = x2 + 5x (iii) g(x) = x2 + 3x − 10 (iv) g(x) = x2 − 4x + 4 (v) g(x) = x2 − 2x − 6 (vi) g(x) = x2 + 3x + 4 (b) For each function in part (a), use a graphing utility to graph f(x) = (x − 2) ( g(x).
One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2, 4) on the graph of the quadratic function f(x) = x2, as shown in the figure.(a) Find the slope m1 of the line joining (2, 4) and (3, 9). (b)
A rancher plans to fence a rectangular pasture adjacent to a river (see figure). The rancher has 100 meters of fencing, and no fencing is needed along the river.(a) Write the area A of the pasture as a function of x, the length of the side parallel to the river. What is the domain of A? (b) Graph
A wire 100 centimeters in length is cut into two pieces. One piece is bent to form a square and the other to form a circle. Let x equal the length of the wire used to form the square.Figure(a) Write the function that represents the combined area of the two figures. (b) Determine the domain of the
At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the
(a) Find the zeros of each quadratic function g(x). (i) g(x) = 2x2 + 5x − 3 (ii) g(x) = − x2 − 3x − 2 (b) For each function in part (a), find the zeros of f(x) = g(1/2 x). (c) Describe the connection between the results in parts(a) and (b)?
Quonset huts were developed during World War II. They were temporary housing structures that could be assembled quickly and easily. A Quonset hut is shaped like a half cylinder. A manufacturer has 600 square feet of material with which to build a Quonset hut. (a) The formula for the surface area of
Show that if f (x) = ax3 + bx2 + cx + d, then f(k) = r, where r = ak3 + bk2 + ck + d, using long division. In other words, verify the Remainder Theorem for a third-degree polynomial function?
In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of y3 + y2. To be able to use this table, the Babylonians sometimes used the method below to manipulate the equation.Then they would find (a2c)/b3 in the y3 + y2 column of the
Can a cubic function with real coefficients have two real zeros and one complex zero? Explain.
(a) Complete the table.(b) Use the table to make a conjecture relating the sum of the zeros of a polynomial function to the coefficients of the polynomial function. (c) Use the table to make a conjecture relating the product of the zeros of a polynomial function to the coefficients of the
Determine whether the statement is true or false. If false, provide one or more reasons why the statement is false and correct the statement. Letf x) = ax3 + bx2 + cx + d, a ( 0And let f (2) = ˆ’1. Then
The parabola shown in the figure has an equation of the form y = ax2 + bx + c. Find the equation of this parabola using each method.(a) Find the equation analytically.(b) Use the regression feature of a graphing utility to find the equation.
Is it possible for the graph of a rational function to have all three types of asymptotes? Why or why not? To work an extended application analyzing the total numbers of military personnel on active duty from 1984 through 2014, visit this text's website at LarsonPrecalculus.com.
1. A ________ is the intersection of a plane and a double-napped cone. 2. The equation (x − h)2 + (y − k)2 = r2 is the standard form of the equation of a ________ with center ________ and radius ________. 3. A ________ is the set of all points (x, y) in a plane that are equidistant from a fixed
Find the focus and directrix of the parabola. Then sketch the parabola. 1. y = 12x2 2. y = −4x2 3. y2 = −6x 4. y2 = 3x
Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 1. Focus: (3, 0) 2. Focus: (0, 1/2) 3. Directrix: y = 2 4. Directrix: x = −4
Find the standard form of the equation of the parabola and determine the coordinates of the focus.1.2.
1. The light bulb in a flashlight is at the focus of the parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation for a cross section of the flashlight's reflector with its focus on the positive x-axis and its vertex at the origin.Figure2. The receiver
Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway.The cables touch the roadway at the midpoint between the towers.(a) Sketch the bridge on a rectangular coordinate
A simply supported beam (see figure) is 64 feet long and has a load at the center. The deflection of the beam at its center is 1 inch. The shape of the deflected beam is parabolic.(a) Write an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.1.2. 3. Vertices: (±5, 0); foci: (±2, 0) 4. Vertices: (0, ±8); foci: (0, ±4)
Find the vertices and eccentricity of the ellipse. Then sketch the ellipse.1.2. 3. 4.
Find the standard form of the equation of the ellipse with the given vertices, eccentricity e, and center at the origin. 1. Vertices: ((5, 0); e = 4/5 2. Vertices: (0, (8); e = 1/2
A mason is building a semielliptical fireplace arch that has a height of 2 feet at the center and a width of 6 feet along the base (see figure). The mason draws the semi ellipse on the wall by the method shown in Figure 4.22 on page 330. Find the positions of the thumbtacks and the length of the
A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Sketch the arch of the tunnel on a rectangular coordinate system with the center of the road entering the tunnel at the origin. Label the coordinates of
Repeat Exercise 56 for a semielliptical arch with a major axis of 40 feet and a height at the center of 15 feet. The dimensions of the truck are 10 feet wide by 14 feet high. Refer to Exercise 56, A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet
A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. An ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the
Sketch the ellipse using the latera recta (see Exercise 58).1.2. 3. 9x2 + 4y2 = 36 4. 3x2 + 6y2 = 30
Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 1. Vertices: (0, ±2); foci: (0, ±6) 2. Vertices: (±4, 0); foci: (±5, 0) 3. Vertices: (±1, 0); asymptotes: y = ±3x 4. Vertices: (0, ±3); asymptotes: y = ±3x
Find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid.1.2. 3. 4.
A cross section of a sculpture can be modeled by a hyperbola (see figure).(a) Write an equation that models the curved sides of the sculpture. (b) Each unit on the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 18 feet?
A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at focus A is reflected to focus B. Find the vertex of the mirror when its mount at the top edge of the mirror has coordinates (24, 24).
When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, then the sound waves intersect the ground in a hyperbola with the airplane directly above its center, and a sonic boom is heard along the
Long-distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a
1. The equation x2 − y2 = 144 represents a circle. 2. The major axis of the ellipse y2 + 16x2 = 64 is vertical? Determine whether the statement is true or false. Justify your answer.
1. It is possible for a parabola to intersect its directrix? 2. When the vertex and focus of a parabola are on a horizontal line, the directrix of the parabola is vertical?
Consider the ellipse(a) The area of the ellipse is given by A = ab. Write the area of the ellipse as a function of a. (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a), and make a conjecture about the shape of the
Match the equation with its graph. [The graphs are labeled (a)-(f).](a)(b) (c) (d) (e) (f) 1. x2 = - 2y 2. y2 = 2x 3. x2 / y + y2 = 1 4. x2 - y2 / y = 1
In parts (a)-(d), describe how a plane could intersect the double-napped cone to form each conic section (see figure).(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
Explain how to use a graphing utility to check your graph in Exercise 43. What equation(s) would you enter into the graphing utility?Refer in Exercise 43,
1. How can you tell whether an ellipse is a circle from the equation? 2. Is the graph of x2 − 4y4 = 4 a hyperbola? Explain.
1. The graph of x2 − y2 = 0 is a degenerate conic. Sketch this graph and identify the degenerate conic? 2. Which part of the graph of the ellipse 4x2 + 9y2 = 36 does each equation represent? Answer without graphing the equations. (a) x = - 3 / 2 (4 - y2 (b) y = 2 / 3 (9 - x2
1. Write a paragraph discussing the changes in the shape and orientation of the graph of the ellipseas a increases from 1 to 8. 2. Use two thumbtacks, a string, and a pencil to draw an ellipse, as shown in Figure 4.22 on page 330. Vary the length of the string and the distance between the
Use the definition of an ellipse to derive the standard form of the equation of an ellipse. (The sum of the distances from a point (x, y) to the foci is 2a.)
Use the definition of a hyperbola to derive the standard form of the equation of a hyperbola. (The absolute value of the difference of the distances from a point (x, y) to the foci is 2a.)
Match the description of the conic with its standard equation. The equations are labeled (a)-(f). a. (x - h)2 / a2 + (y - 5)2 / b2 = 1 b. (x - h)2 / a2 - (y - k)2 / b2 = 1 c. (y - 5)2 / a2 - (x - h)2 / b2 = 1 d. (x - h)2 / b2 + (y - k)2 / a2 = 1 e. (x - h)2 = 4p(y - 5) f. (y - 5)2 = 4p(x -
In Exercises 1-4, find the center and radius of the circle. 1. x2 + y2 = 49 2. x2 + y2 = 1 3. (x − 4)2 + (y − 5)2 = 36 4. (x + 8)2 + (y + 1)2 = 144
In Exercises 1-4, write the equation of the circle in standard form, and then find its center and radius. 1. x2 + y2 − 8y = 0 2. x2 + y2 − 10x + 16 = 0 3. x2 + y2 − 2x + 6y + 9 = 0 4. 2x2 + 2y2 − 2x − 2y − 7 = 0
In Exercises 1-4, find the vertex, focus, and Directix of the parabola. Then sketch the parabola. 1. (x − 1)2 + 8(y + 2) = 0 2. (x + 2) + (y − 4)2 = 0 3. (y + 1 / 2)2 = 2(x − 5) 4. (x + 1 / 2)2 = 4(y − 3)
In Exercises 1-4, find the standard form of the equation of the parabola with the given characteristics. 1. Vertex: (3, 2); Focus: (1, 2) 2. Vertex: (-1, 2); Focus: (-1, 2) 3. Vertex: (0, 4); Directrix: y = 2 4. Vertex: (2, 1); Directrix: x = 1
A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour (see figure). When this velocity is multiplied by ˆš2, the satellite has the minimum velocity necessary to escape Earth's gravity and follow a parabolic path with the center of
Water flowing from a horizontal pipe 48 feet above the ground has the shape of a parabola whose vertex (0, 48) is at the end of the pipe (seefigure). The water strikes the ocean at the point (10 ˆš 3, 0).(a) Find the standard form of the equation that represents the path of the water. (b)
A ball is thrown from the top of a 100-foot tower with a velocity of 28 feet per second. (a) Find the equation that represents the parabolic path. (b) How far does the ball travel horizontally before it strikes the ground?
A cargo plane is flying at an altitude of 500 feet and a speed of 135 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel will the crate travel horizontally before it hits the ground?
In Exercises 1-4, find the center, foci, and vertices of the ellipse. Then sketch the ellipse 1. (x - 1)2 / 9 + (y - 5)2 / 25 = 1 2. (x - 6)2 / 4 + (y + 7)2 / 16 = 1 3. (x + 2)2 + (y + 4)2 / 1/4 = 1 4. (x - 3)2 / 25/9 + (y - 8)2 = 1
In Exercises 1-4, find the standard form of the equation of the ellipse with the given characteristics. 1. Vertices: (3, −3), (3, 3); minor axis of length 2 2. Vertices: (−2, 3), (6, 3); minor axis of length 6 3. Foci: (0, 0), (4, 0); major axis of length 8 4. Foci: (0, 0), (0, 8); major axis
The dwarf planet Pluto moves in an elliptical orbit with the sun at one of the foci, as shown in the figure. The length of half of the major axis, a, is 3.67 Ã— 109 miles, and the eccentricity is 0.249. Find the least distance (perihelion) and the greatest distance (aphelion) of Pluto
In Australia, football by Australian Rules is played on elliptical fields. The field can be a maximum of 155 meters wide and a maximum of 185 meters long. Let the center of a field of maximum size be represented by the point (0, 77.5). Find the standard form of the equation that represents this
In Exercises 1-2, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid 1. (x - 2)2 / 16 - (y + 1)2 / 9 = 1 2. (x - 1)2 / 144 (7 - 4)2 / 25 = 1
In Exercises 1-4, identify the conic. Then describe the translation of the conic from standard position.1. (x + 2)2 + (y - 1)2 = 42. (y ˆ’ 1)2 = 4(2)(x + 2) 3. (y + 3)2 / 4 - (x -1)2 = 1 4. (x - 2)2 / 9 + (y + 1)2 / 4 = 1
In Exercises 1-4, find the standard form of the equation of the hyperbola with the given characteristics. 1. Vertices: (0, 2), (0, 0); foci: (0, 3), (0, −1) 2. Vertices: (1, 2), (5, 2); foci: (0, 2), (6, 2) 3. Foci: (−8, 5), (0, 5); transverse axis of length 4 4. Foci: (7, −5), (7, 3);
In Exercises 1-4, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation from standard position. 1. y2 - x2 + y = 0 2. x2 + y2 - 6x + 4y + 9 = 0 3. 16y2 + 128x + 8y - 7 = 0 4. 4x2 y2 4x 3 = 0
In Exercises 1-3, determine whether the statement is true or false. Justify your answer. 1. The conic represented by the equation 3x2 + 2y2 − 18x − 16y + 58 = 0 is an ellipse. 2. The graphs of x2 + 10y − 10x + 5 = 0 and x2 + 16y2 + 10x − 32y − 23 = 0 do not intersect. 3. A hyperbola can
Consider the ellipse (x2 / a2) + (y2 / b2) = 1.a. Show that the equation of the ellipse can be written as (xh)2 / a2 + (yk)2 / a2(1 - e2) = 1 Where e is the eccentricity.b. Use a graphing utility to graph the ellipse (x - 2)2 / 4 + (y - 3)2 / 4(1 - e2) = 1 for e = 0.95, 0.75, 0.5, 0.25, and 0. Make
Find the domain of the function and discuss the behavior of f near any excluded x-values. 1. f(x) = 3x / x+10 2. f(x) = 4x3 / 2 + 5x 3. f(x) = 8 / x2-10x+24 4. f(x) = x2+ x - 2 / x2 -4x +4
1. The cost C (in dollars) of producing x units of a product is given by C = 0.5x + 500 and the average cost per unit C̅ is given by C̅ = C / x = 0.5x + 500 / x, x>0. Determine the average cost per unit as x increases without bound. 2. The cost C (in millions of dollars) for the federal
(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 1. f(x) = -3/2x2 2. f(x) = 4 / x 3. g(x) = 2 + x / 1 - x
(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 1. f(x) = 2x3 / x2 + 1 2. f(x) = x2 + 1 / x + 1 3. f(x) = x2 + 3x - 10 / x + 2
The cost C (in dollars) of producing x units of a product is given by C = 100,000 + 0.9x and the average cost per unit C̅ is given by C̅ = C / x = 100,000 + 0.9x / x, x>0. (a) Sketch the graph of the average cost function. (b) Find the average costs when x = 1000, x = 10,000, and x = 100,000. (c)
A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom margins are each 2 inches deep and the margins on each side are 2 inches wide. (a) Show that the total area A of the page is A = 2x (2x + 7) / x - 4 (b) Determine the domain of the function based
A parks and wildlife commission releases 80,000 fish into a lake. After t years, the population years, the population N of the fish (in thousands) is given by N = 20(3t +4) / 0.05t +1, t ≥ 0. (a) Sketch the graph of the function. (b) Find the population when t = 5, t = 10, and t = 25. (c) What is
1. The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is hours after injection into muscle tissue is given by C (t) = (2t + 1) / (t2 + 4), t > 0. (a) Determine the horizontal asymptote of the graph of the function and interpret its meaning in the context
Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 1. Focus: (−6, 0) 2. Focus: (0, 7) 3. Directrix: y = −3 4. Directrix: x = 3 5. Passes through the point (3, 6); horizontal axis 6. Passes through the point (4, −2); vertical axis
1. A cross section of a large parabolic satellite dish is modeled by y = x2 / 200, ˆ’100 ‰¤ x ‰¤ 100 (see figure). The receiving and transmitting equipment is positioned at the focus. Find the coordinates of the focus.2. Each cable of a suspension bridge is
Find all vertical and horizontal asymptotes of the graph of the function. 1. f (x) = 6x2 / x + 3 2. f (x) = 2x2 + 5x -3 / x2 + 2 3. g (x) = x2 / x2 - 4 4. g (x) = x + 1 / x2 - 1 5. h (x) = 5x + 20 / x2 - 2x - 24 6. h (x) = x3 - 4x2 / x2 + 3x + 2
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.1.2. 3. Vertices: (0, ±7); foci: (0, ±6)
1. A semielliptical archway is formed over the entrance to an estate. The arch is set on pillars that are 10 feet apart and has a height (atop the pillars) of 4 feet (see figure). Describe the location of the foci.2. You are building a wading pool that is in the shape of an ellipse. An equation for
Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 1. Vertices: (0, ±1); foci: (0, ±5) 2. Vertices: (±4, 0); foci: (±6, 0) 3. Vertices: (±1, 0); asymptotes: y = ±2x 4. Vertices: (0, ±2); asymptotes: y = ± 2/ √5 x
Find the standard form of the equation of the parabola with the given characteristics. 1. Vertex: (4, 2); focus: (4, 0) 2. Vertex: (2, 0); focus: (0, 0) 3. Vertex: (8, −8); directrix: x = 1 4. Focus: (5, 6); directrix: y = 0
Find the standard form of the equation of the ellipse with the given characteristics. 1. Vertices: (0, 2), (4, 2); minor axis of length 2 2. Vertices: (5, 0), (5, 12); minor axis of length 10 3. Vertices: (2, −2), (2, 8); foci: (2, 0), (2, 6)
Find the standard form of the Equation of the hyperbola with the given characteristics. 1. Vertices: (−10, 3), (6, 3); foci: (−12, 3), (8, 3) 2. Vertices: (2, −2), (2, 2); foci: (2, −4), (2, 4) 3. Vertices: (3, −4), (3, 4); passes through the point (4, 6) 4. Vertices: (±6,
Identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation from standard position. 1. x2 − 6x + 2y + 9 = 0 2. y2 − 12y − 8x + 20 = 0 3. x2 + y2 − 2x − 4y + 1 = 0
1. A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level?2. A church window is bounded above by a parabola and below by the arc of a circle (see figure). (a) Find equations for the

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