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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. 1. [−2, 6) 2. (−7, 4) 3. [−1, 5] 4. (2, 10] 5. (11,∞) 6. [−5, ∞) 7. (-∞, −2) 8. (-∞, 7]
Use a graphing utility to graph the inequality and identify the solution set. 1. 7x > 21 2. −4x ≤ 9 3. 8 - 3x ≥ 2 4. 20 < 6x - 1
Use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. Equation 1. y = 3x - 1 Inequalities (a) y ≥ 2 (b) y ≤ 0 Equation 2. y = 2 / 3 x + 1 Inequalities (a) y ≤ 5 (b) y ≥0
1. The graph of ∣x − 5∣ < 3 can be described as all real numbers less than three units from 5. Give a similar description of ∣x − 10∣ < 8.2. The graph of ∣x − 2∣ > 5 can be described as all real numbers more than five units from 2. Give a similar description of ∣x -
Use absolute value notation to define the interval (or pair of intervals) on the real number line.1.2. 3. 4. 5. All real numbers at least three units from 7 6. All real numbers more than five units from 8 7. All real numbers less than four units from ˆ’3 8. All real numbers no more than
Write an inequality to describe the situation.1. During a trading day, the price P of a stock is no less than $7.25 and no more than $7.75.2. During a month, a person's weight w is greater than 180 pounds but less than 185.5 pounds.3. The expected return r on an investment is no more than 8%.4.
Determine the interval in which the person's heart rate is from 50% to 85% of the maximum heart rate. 1. A 20-year-old 2. A 40-year-old
1. You are considering two job offers. The first job pays $13.50 per hour. The second job pays $9.00 per hour plus $0.75 per unit produced per hour. How many units must you produce per hour for the second job to pay more per hour than the first job? 2. You are considering two job offers. The first
1. The revenue from selling x units of a product is R = 115.95x. The cost of producing x units is C = 95x + 750. To obtain a profit, the revenue must be greater than the cost. For what values of x does this product return a profit? 2. The revenue from selling x units of a product is R = 24.55x. The
1. A doughnut shop sells a dozen doughnuts for $7.95. Beyond the fixed costs (rent, utilities, and insurance) of $165 per day, it costs $1.45 for enough materials and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies between $400 and $1200. Between what levels (in
1. An equation that relates the college grade-point averages y and high school grade-point averages x of the students at a college is y = 0.692x + 0.988.(a) Use a graphing utility to graph the model.(b) Use the graph to estimate the values of x that predict a college grade-point average of at least
1. Between two consecutive zeros, a polynomial must be entirely ________ or entirely ________. 2. To solve a polynomial inequality, find the ________ numbers of the inequality, and use these numbers to create ________ ________ for the inequality. 3. A rational expression can change sign at its
In Exercises 1-2, solve the inequality. Then graph the solution set. 1. 2x2 + 4x < 0 2. 3x2 - 9x ≥ 0
In Exercises 1-2, explain what is unusual about the solution set of the inequality. 1. 4x2 − 4x + 1 ≤ 0 2. x2 + 3x + 8 > 0
In Exercises 1-2, solve the inequality. Then graph the solution set. 1. 4x - / x > 0 2. x2 - 1 / x < 0
In Exercises 1-2, determine whether each value of x is a solution of the inequality. 1. x2 - 3 < 0 a. x = 3 b. x = 0 c. x = 3/2 d. x = -5 2. Inequality x2 - 2x - 8 ≥ 0 Values a. x = -2 b. x = 0 c. x = -4 d. x = 1
In Exercises 1-4, use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. Equation 1. y = −x2 + 2x + 3 Inequalities a. y ≤ 0 b. y ≥ 3 2. y = 1/2x2 - 2x + 1 a. y ≤ 0 b. y ≥ 7 3. y = 1/8x3 - 1/2x a. y ≥ 0 b. y ≤ 0
In Exercises 1-4, solve the inequality. (Round your answers to two decimal places) 1. 0.3x2 + 6.26 < 10.8 2. -1.3x2 + 3.78 > 2.12 3. −0.5x2 + 12.5x + 1.6 > 0 4. 1.2x2 + 4.8x + 3.1 < 5.3
In below Exercises, use the position equationwhere s represents the height of an object (in feet), v0 represents the initial velocity of the object (in feet per second), s0 represents the initial height of the object (in feet), and t represents the time (in seconds). A projectile is fired straight
In below Exercises, use the position equationwhere s represents the height of an object (in feet), v0 represents the initial velocity of the object (in feet per second), s0 represents the initial height of the object (in feet), and t represents the time (in seconds). A projectile is fired straight
The revenue and cost equations for a product are R = x(75 − 0.0005x) and C = 30x + 250,000, where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to obtain a profit of at least $750,000? What is the price per unit?
The revenue and cost equations for a product are R = x(50 − 0.0002x) and C = 12x + 150,000, where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to obtain a profit of at least $1,650,000? What is the price per unit?
In Exercises 1-2, find the domain of the expression. Use a graphing utility to verify your result. 1. √4 - x2 2. √x2 - 9
The table shows the numbers N (in millions) of students enrolled in elementary and secondary schools in the United States from 2005 through 2014.(a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t = 5 corresponding to 2005. (b) Use the regression
The maximum safe load uniformly distributed over a one-foot section of a two-inch-wide wooden beam can be approximated by the model Load = 168.5d2 − 472.1 where d is the depth of the beam. (a) Evaluate the model for d = 4, d = 6, d = 8, d = 10, and d = 12. Use the results to create a bar
A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?
A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie?
The table shows the mean salaries S (in thousands of dollars) of public school classroom teachers in the United States from 2002 through 2013.A model that approximates these data is S = 40.32 + 3.53t / 1 + 0.039t, 2 ‰¤ t ‰¤ 13 where t represents the year, with t = 2
The solution set of the inequality 3 / 2 x2 + 3x + 6 ≥ 0 is the entire set of real numbers.
Use a graphing utility to verify the results in Example 4. For instance, the graph of y = x2 + 2x + 4 is shown below. Notice that the y-values are greater than 0 for all values of x, as stated in Example 4(a). Use the graphing utility to graph y = x2 + 2x + 1, y = x2 + 3x + 5, and y = x2
In Exercises 1-2, (a) Find the interval(s) for b such that the equation has at least one real solution and (b) Write a conjecture about the interval(s) based on the values of the coefficients.1. x2 + bx + 9 = 02. x2 + bx - 9 = 0
In Exercises 1-2, find the zeros of the expression. 1. x2 - 3x - 18 2. 9x3 - 25x2
1. The simplest mathematical model for relating two variables is the ________ equation in two variables y = mx + b. 2. For a line, the ratio of the change in y to the change in x is the ________ of the line. 3. The ________-________ form of the equation of a line with slope m passing through the
The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width x surrounds the garden.(a) Draw a diagram that gives a visual representation of the problem.(b) Write the equation for the perimeter y of the walkway in terms of x.(c) Use a graphing utility
In Exercises 1 and 2, determine whether the statement is true or false. Justify your answer. 1. A line with a slope of −5/7 is steeper than a line with a slope of −6/7. 2. The line through (−8, 2) and (−1, 4) and the line through (0, −4) and (−7, 7) are parallel.
Explain how you can use slope to show that the points A(−1, 5), B(3, 7), and C(5, 3) are the vertices of a right triangle.
Explain why the slope of a vertical line is undefined.
Describe the error.Line b has a greater slope than line a.
Find d1 and d2 in terms of m1 and m2, respectively (see figure). Then use the Pythagorean Theorem to find a relationship between m1 and m2.
The slopes of two lines are −4 and 5/2. Which is steeper? Explain.
Use a graphing utility to compare the slopes of the lines y = mx, where m = 0.5, 1, 2, and 4. Which line rises most quickly? Now, let m = −0.5, −1, −2, and −4. Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope
In below Exercises 1 and 2, sketch the lines through the point with the given slopes on the same set of coordinate axes. Point 1. (2,3) Slopes a. 0 b. 1 c. 2 d. -3 2. (-4, 1) a. 3 b. -3 c. 1/2 d. Undefined
In Exercises 1-2, find a relationship between x and y such that x, y is equidistant (the same distance) from the two points. 1. (4, −1), (−2, 3) 2. (6, 5), (1, −8)
In below Exercises, estimate the slope of the line.1.2.
In below Exercises, find the slope and y-intercept (if possible) of the line. Sketch the line. 1. y = 5x + 3 2. y = -x 10 3. y = 3/4x -1 4. y = 2/3x + 2
In Exercises 1-4, find the slope of the line passing through the pair of points. 1. (0, 9), (6,0) 2. (10,0), (0,-5) 3. (-3, -2),(1,6) 4. (2, −1), (−2, 1)
In Exercises 1-4, use the slope of the line and the point on the line to find three additional points through which the line passes. (There are many correct answers.) 1. m = 0, (5, 7) 2. m = 0, (3, −2) 3. m = 2, (-5,4) 4. m = - 2, (0, -9)
In Exercises 1-4, find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line. 1. m = 3, (0, −2) 2. m = −1, (0, 10) 3. m = −2, (−3, 6) 4. m = 4, (0, 0)
In Exercises 1-4, find an equation of the line passing through the pair of points. Sketch the line. 1. (5, −1), (−5, 5) 2. (4, 3), (−4, −4) 3. (−7, 2), (−7, 5) 4. (−6, −3), (2, −3)
In Exercises 1-4, determine whether the lines are parallel, perpendicular, or neither. 1. L1 : y = -2/3x -3 L2 : y = -2/3x + 4 2. L1 : = y 1/4x - 1 L2 : y = 4x + 7 3. L1 : y 1/2x -3 L2 : y = -1/2x +1 4. L1 : y = 4/5x - 5 L2 : y = 5/4x + 1
In Exercises 1-4, determine whether the lines L1 and L2 passing through the pairs of points are parallel, perpendicular, or neither.1. L1: (0, −1), (5, 9) L2: (0, 3), (4, 1)2. L1: (−2, −1), (1, 5)L2: (1, 3), (5, −5)3. L1: (−6, −3), (2, −3) L2: (3, - 1/2),(6, - 1/2)4. L1:
In Exercises 1-2, find equations of the lines that pass through the given point and are(a) Parallel to and(b) Perpendicular to the given line.1. 4x − 2y = 3, (2, 1)2. x + y = 7, (−3, 2)
In Exercises 1-4, use the intercept form to find the general form of the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts (a, 0) and (0, b) is x / a + y / b = 1, a ≠ 0, b ≠ 0. 1. x-intercept: (3, 0) y-intercept: (0, 5) 2.
The slopes of lines representing annual sales y in terms of time x in years are given below. Use the slopes to interpret any change in annual sales for a one-year increase in time.(a) The line has a slope of m = 135.(b) The line has a slope of m = 0.(c) The line has a slope of m = −40.
The graph shows the sales (in billions of dollars) for Apple Inc. in the years 2009 through 2015.(a) Use the slopes of the line segments to determine the years in which the sales showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting the points for
In below Exercises, identify the line that has each slope.1.a. m = 2/3b. m is undefined.c. m = -22. a. m = 0 b. m = -3/4 c. m = 1
From the top of a mountain road, a surveyor takes several horizontal measurements x and several vertical measurements y, as shown in the table (x and y are measured in feet).(a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find
In Exercises 1 and 2, you are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 16
A sandwich shop purchases a used pizza oven for $875. After 5 years, the oven will have to be discarded and replaced. Write a linear equation giving the value V of the equipment during the 5 years it will be in use.
A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use.
Write a linear equation that expresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. Use the fact that water freezes at 0°C (32°F) and boils at 100°C (212°F)
The average weight of a male child's brain is 970 grams at age 1 and 1270 grams at age 3.(a) Assuming that the relationship between brain weight y and age t is linear, write a linear model for the data.(b) What is the slope and what does it tell you about brain weight?(c) Use your model to estimate
A roofing contractor purchases a shingle delivery truck with a shingle elevator for $42,000. The vehicle requires an average expenditure of $9.50 per hour for fuel and maintenance, and the operator is paid $11.50 per hour.(a) Write a linear equation giving the total cost C of operating this
As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You receive an offer for a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. (b) Write a linear
You are in a boat 2 miles from the nearest point on the coast (see figure). You plan to travel to point Q, 3 miles down the coast and 1 mile inland. You row at 2 miles per hour and walk at 4 miles per hour.(a) Write the total time T (in hours) of the trip as a function of the distance x (in
The Heaviside function:is widely used in engineering applications. (See figure.) To print an enlarged copy of the graph, go to MathGraphs.com. Sketch the graph of each function by hand. (a) H(x) ˆ’ 2 (b) H(x ˆ’ 2) (c) ˆ’H(x) (d) H (ˆ’x) (e) 1/2 H(x) (f)
Let f (x) = 1 / (1 - x) (a) Find the domain and range of f. (b) Find f {f (x)}. What is the domain of this function? (c) Find f {f (f (x)}. Is the graph a line? Why or why not?
Show that the Associative Property holds for compositions of functions-that is, (f o (g o h)) (x) = {(f o g) o h} (x).
Use the graph of the function f to sketch the graph of each function. To print an enlarged copy of the graph, go to MathGraphs.com.(a) f (x + 1) (b) f (x) + 1 (c) 2f (x) (d) f (ˆ’x) (e) ˆ’f (x) (f) |f (x)| (g) f (|x|)
Use the graphs of f and f ˆ’1 to complete each table of function values.(a) (b) (c) (d)
For the numbers 2 through 9 on a cellphone keypad (see figure), consider two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain.
What can be said about the sum and difference of each pair of functions? (a) Two even functions (b) Two odd functions (c) An odd function and an even function
The functions f (x) = x and g(x) = −x are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a formula for a family of linear functions that are their own inverse functions.
Prove that a function of the form y = a2nx2n + a2n-2x2n-2 + .... + a2x2 + ao is an even function.
A golfer is trying to make a hole-in-one on the miniature golf green shown. The golf ball is at the point (2.5, 2) and the hole is at the point (9.5, 2). The golfer wants to bank the ball off the side wall of the green at the point (x, y). Find the coordinates of the point (x, y). Then write an
At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400-mile trip. (a) What was the total duration of the voyage in hours? (b)
Consider the function f (x) = −x2 + 4x − 3. Find the average rate of change of the function from x1 to x2. (a) x1 = 1, x2 = 2 (b) x1 = 1, x2 = 1.5 (c) x1 = 1, x2 = 1.25 (d) x1 = 1, x2 = 1.125 (e) x1 = 1, x2 = 1.0625 (f) Does the average rate of change seem to be approaching one value? If so,
Consider the functions f (x) = 4x and g(x) = x + 6. (a) Find (f o g) (x). (b) Find (f o g) 1(x). (c) Find f −1(x) and g−1(x). (d) Find (g−1 o f−1) (x) and compare the result with that of part (b). (e) Repeat parts (a) through (d) for f (x) = x3 + 1 and g(x) = 2x. (f) Write two one-to-one
Determine whether the equation represents y as a function of x? 1. x2 + y2 = 4 2. x2 − y = 9 3. y = (16 − x2 4. y = (x + 5
Find each function value, if possible? 1. f (x) = 3x − 5 (a) f (1) (b) f (−3) (c) f (x + 2) 2. V(r) = 4/3(r3 (a) V(V3) (b) V(3/2) (c) V(2r) 3. g(t) = 4t2 − 3t + 5 (a) g(2) (b) g(t − 2) (c) g(t) − g(2) 4. h(t) = −t2 + t + 1 (a) h(2) (b) h(−1) (c) h(x + 1)
Complete the table.1. f(x) = ˆ’ x2 + 52. h(t) = ½ |t + 3| 3. 4.
Find all real values of x for which f(x) = 0. 1. f(x) = 15 − 3x 2. f(x) = 4x + 6 3. f(x) = 3x - 4/5 4. f(x) = 12 − x2/8
Find the value(s) of x for which f(x) = g(x). 1. f(x) = x2, g(x) = x + 2 2. f (x) = x2 + 2x + 1, g(x) = 5x + 19 3. f(x) = x4 − 2x2, g(x) = 2x2 4. f(x) = (x − 4, g(x) = 2 − x
Find the domain of the function. 1. f(x) = 5x2 + 2x − 1 2. g(x) = 1 − 2x2 3. g(y) = (y + 6 4. f(t) = 3(t + 4
Determine whether the relation represents y as a function of x.1. Domain, x Range, y2. Domain, x Range, y 3. 4.
An open box of maximum volume is made from a square piece of material 24centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure).(a) The table shows the volumes V (in cubic centimeters) of the box for various heights x (in centimeters). Use the table to
The cost per unit in the production of an MP3 player is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by $0.15 per MP3 player for each unit ordered in excess of 100 (for example, the charge is reduced to $87 per
1. Write the area A of a square as a function of its perimeter P? 2. Write the area A of a circle as a function of its circumference C?
You throw a baseball to a child 25 feet away. The height y (in feet) of the baseball is given by y = - 1/10x2 + 3x + 6 Where x is the horizontal distance (in feet) from where you threw the ball. Can the child catch the baseball while holding a baseball glove at a height of 5feet?
A rectangular package has a combined length and girth (perimeter of a cross section) of 108 inches (see figure).(a) Write the volume V of the package as a function of x. What is the domain of the function? (b) Use a graphing utility to graph the function. Be sure to use an appropriate window
A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (2, 1) (see figure). Write the area A of the triangle as a function of x, and determine the domain of the function.
1. A rectangle is bounded by the x-axis and the semicircle y = (36 ˆ’ x2 (see figure). Write the area A of the rectangle as a function of x, and graphically determine the domain of the function.2. The percent p of prescriptions filled with generic drugs at CVS Pharmacies from 2008 through
The median sale price p (in thousands of dollars) of an existing one-family home in the United States from 2002 through 2014 (see figure) can be approximated by the modelWhere t represents the year, with t = 2 corresponding to 2002. Use this model to find the median sale price of an existing
A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let x be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost C as
The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6000. The inventor sells each game for $1.69. Let x be the number of games produced. (a) The total cost for a business is the sum of the variable cost and the fixed costs.
A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let h represent the height of the balloon and let d represent the distance between the balloon and the receiving station. (b)
The function F(y) = 149.76(10y5)2 estimates the force F (in tons) of water against the face of a dam, where y is the depth of the water (in feet).(a) Complete the table. What can you conclude from the table?(b) Use the table to approximate the depth at which the force against the dam is 1,000,000
For groups of 80 or more people, a charter bus company determines the rate per person according to the formulaRate = 8 ˆ’ 0.05(n ˆ’ 80), n ( 80Where the rate is given in dollars and n is the number of people.(a) Write the revenue R for the bus company as a function of n.(b) Use
The table shows the numbers of tax returns (in millions) made through e-file from 2007 through 2014. Let f (t) represent the number of tax returns made through e-file in the year t.Year __________ Number of Tax Returns.........................Made Through E-File2007
Find the difference quotient and simplify your answer.1. f(x) = x2 - 2x + 4,h ( 0 2. f(x) = 5x - x2, h ( 0 3. f(x) = x3 + 3x, h ( 0 4. f (x) = 4x3 ˆ’ 2x, h ( 0

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