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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

A forester is making a gasoline-oil mixture for a chainsaw engine. The forester has 2 gallons of a mixture that is 32 parts gasoline and 1 part oil. How many gallons of gasoline should the forester add to bring the mixture to 50 parts gasoline and 1 part oil?
A grocer mixes peanuts that cost $1.49 per pound and walnuts that cost $2.69 per pound to make 100 pounds of a mixture that costs $2.21 per pound. How much of each kind of nut is in the mixture?
1. Solve for h: A = 1/2 bh.2. Solve for l: V = lwh.
1. Solve for C: S = C + RC. 2. Solve for L: S = L − RL.
1. Solve for r: A = P + Prt. 2. Solve for b: A = 1/2(a + b)h.
The average body temperature of a person is 98.6°F. What is this temperature in degrees Celsius?
The average body temperature of a dog is 101.3°F. What is this temperature in degrees Celsius?
The melting point of francium is 27°C. What is this temperature in degrees Fahrenheit?
The boiling point of bromine is 58.8°C. What is this temperature in degrees Fahrenheit?
1. A billiard ball has a volume of 5.96 cubic inches. Find the radius of the billiard ball. 2. The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank.
You have a uniform beam of length L with a fulcrum x feet from one end (see figure). Objects with weights W1 and W2 are placed at opposite ends of the beam. The beam balances when W1x = W2(L ˆ’ x).Find x such that each beam will balance. (a) Two children weighing 50 pounds and 75 pounds
To determine a building's height, you measure the shadows cast by the building and a nearby four-foot post (see figure).(a) Write a verbal model for the situation. (b) Translate your verbal model into an algebraic equation.
1. "8 less than z cubed divided by the difference of z squared and 9" can be written as (z3 − 8) / (z − 9)2.2. The expression x3/(x − 4)2 can be described as "x cubed divided by the square of the difference of x and 4".
The area of a circle with a radius of 2 inches is less than the area of a square with a side length of 4 inches.
The volume of a cube with a side length of 9.5 inches is greater than the volume of a sphere with a radius of 5.9 inches.
1. A ________ ________ in x is an equation that can be written in the general form ax2 + bx + c = 0, where a, b, and c are real numbers with a ≠ 0. 2. A quadratic equation in x is also called a ________ ________ equation in x. 3. Four methods for solving quadratic equations are ________,
The floor of a one-story building is 14 feet longer than it is wide (see figure). The building has 1632 square feet of floor space.(a) Write a quadratic equation for the area of the floor in terms of w. (b) Find the length and width of the floor.
A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens (see figure). The gardener wants the enclosed area to be 350 square meters. What dimensions should the gardener use to obtain this area?
You construct an open box with a square base (see figure) from 108 square inches of material. The height of the box is 3 inches. What are the dimensions of the box?
You construct an open box from a square piece of material by cutting four-centimeter squares from the corners and turning up the sides (see figure). The volume of the box is 576 cubic centimeters. Find the dimensions of the square piece of material that you use to construct the box.
An above-ground swimming pool contains 1024 cubic feet of water. The rectangular base of the pool is (x + 1) feet by x feet. The height of the water is 4feet.(a) What are the possible dimensions of the base of the pool?(b) If one cubic foot of water weighs approximately 62.4pounds, what
A rectangular classroom seats 72 students. When the seats are rearranged with three more seats in each row, the classroom has two fewer rows. Find the original number of seats in each row.
You drop a coin from the top of the Eiffel Tower in Paris. The building has a height of 984 feet.(a) Use the position equation to write a mathematical model for the height of the coin.(b) Find the height of the coin after 4 seconds.(c) How long does it take the coin to strike the ground?Use the
You drop an object from the top of the CN Tower in Toronto, Ontario. The tower has a height of 1815 feet.(a) Use the position equation to write a mathematical model for the height of the object.(b) Complete the table.(c) From the table in part (b), determine the time interval during which the
An aircraft flying at 550 feet over level terrain drops a supply package.(a) How long does it take the supply package to strike the ground?(b) The aircraft is flying at 138 miles per hour. How far does the supply package travel horizontally during its descent?Use the position equation given in
Some Major League Baseball pitchers can throw a fastball at speeds of up to and over 100 miles per hour. Assume a Major League Baseball pitcher throws a baseball straight up into the air at 100 miles per hour from a height of 6 feet 3 inches.(a) Use the position equation to write a mathematical
The total public debt D (in trillions of dollars) in the United States at the beginning of each year from 2008 through 2014 can be approximated by the modelD = ˆ’ 0.071t2 + 2.94t ˆ’ 10.0, 8 ‰¤ t ‰¤ 14where t represents the year, with t = 8 corresponding
The average ticket price P at movie theaters from 2001 through 2014 can be approximated by the model P = −0.0038t2 + 0.272t + 5.25, 1 ≤ t ≤ 14 where t represents the year, with t = 1 corresponding to 2001. (Source: National Association of Theatre Owners) (a) Use a graphing utility to graph
Doctors treated a patient at an emergency room from 2:00 P.M. to 7:00 P.M. The patient's blood oxygen level L (in percent) during this time period can be modeled byL = ˆ’0.270t2 + 3.59t + 83.1, 2 ‰¤ t ‰¤ 7where t represents the time of day, with t = 2 corresponding
The metabolic rate of an ectothermic organism increases with increasing temperature within a certain range. Experimental data for the oxygen consumption C (in microliters per gram per hour) of a beetle at certain temperatures can be approximated by the modelC = 0.45x2 − 1.65x + 50.75, 10 ≤ x
A winch tows a boat to a dock. The rope is attached to the boat at a point 15 feet below the level of the winch (see figure).(a) Use the Pythagorean Theorem to write an equation giving the relationship between l and x. (b) Find the distance from the boat to the dock when the length l is 75feet.
Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440miles apart. Find the speed of each plane.
1. The quadratic equation −3x2 + x = −5 has two real solutions. 2. If (2x − 3)(x + 5) = 8, then either 2x − 3 = 8 or x + 5 = 8. Determine whether the statement is true or false. Justify your answer.
Write a quadratic equation that has the given solutions. 1. 0 and 4 2. −2 and −8 3. 8 and 14
Solve the equation by extracting square roots. When a solution is irrational, list both the exact solution and its approximation rounded to two decimal places. 1. x2 = 49 2. x2 = 144 3. x2 = 19 4. x2 = 43 5. 3x2 = 81
Solve the quadratic equation by completing the square. 1. x2 + 4x − 32 = 0 2. x2 − 2x − 3 = 0
Rewrite the quadratic portion of the algebraic expression as the sum or difference of two squares by completing the square. 1. 1/x2 - 2x + 5 2. 1/x2 + 6x + 10 3. 4/x2 + 10x + 74
(a) Use a graphing utility to graph the equation,(b) Use the graph to approximate any x-intercepts, (c) Set y = 0 and solve the resulting equation,(d) Compare the result of part(c) with the x-intercepts of the graph.y = (x + 3)2 − 4
Use the discriminant to determine the number of real solutions of the quadratic equation. 1. 9x2 + 12x + 4 = 0 2. x2 + 2x + 4 = 0 3. 2x2 − 5x + 5 = 0 4. −5x2 − 4x + 1 = 0
Use the Quadratic Formula to solve the equation. 1. 2x2 + x − 1 = 0 2. 2x2 − x − 1 = 0 3. 16x2 + 8x − 3 = 0
Solve the quadratic equation by factoring. 1. 6x2 + 3x = 0 2. 8x2 − 2x = 0 3. 3 + 5x − 2x2 = 0
Use the Quadratic Formula to solve the equation. (Round your answer to three decimal places.) 1. 5.1x2 − 1.7x − 3.2 = 0 2. 2x2 − 2.50x − 0.42 = 0 3. −0.67x2 + 0.5x + 1.375 = 0
Solve the equation using any convenient method. 1. x2 − 2x − 1 = 0 2. 14x2 + 42x = 0 3. (x + 2)2 = 64
1. A ________ number has the form a + bi, where a ≠ 0, b = 0. 2. An ________ number has the form a + bi, where a ≠ 0, b ≠ 0. 3. A ________ ________ number has the form a + bi, where a = 0, b ≠ 0. 4. The imaginary unit i is defined as i = ________, where i2 = ________. 5. When a is a
Write the complex number in standard form. 1. 2 + √−25 2. 4 + √−49 3. 1 - √−12 4. 2 - √−18 5. √−40 6. √−27 7. 23 8. 50
Perform the operation and write the result in standard form. 1. (5 + i) + (2 + 3i) 2. (13 − 2i) + (−5 + 6i) 3. (9 − i) − (8 − i) 4. (3 + 2i) − (6 + 13i)
Perform the operation and write the result in standard form. 1. (1 + i) (3 − 2i) 2. (7 − 2i) (3 − 5i) 3. 12i(1 − 9i)
Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. 1. 9 + 2i 2. 8 − 10i 3. −1 - √5i
Write the quotient in standard form. 1. 2 / 4 - 5i 2. 13 / 1 - i 3. 5 + i / 5 - i
Perform the operation and write the result in standard form. 1. 2 / 1 + I - 3/1-i 2. 2i/2 + I + 5/2-i 3. i/3-2i + 2i/3 + 8i
Write the complex number in standard form. 1. √−6 √−2 2. √−5 √−10 3. (√−15)2 4. (√−75)2
Use the Quadratic Formula to solve the quadratic equation. 1. x2 − 2x + 2 = 0 2. x2 + 6x + 10 = 0 3. 4x2 + 16x + 17 = 0
Find real numbers a and b such that the equation is true. 1. a + bi = 9 + 8i 2. a + bi = 10 − 5i 3. (a − 2) + (b + 1)i = 6 + 5i
Simplify the complex number and write it in standard form. 1. −6i3 + i2 2. 4i2 − 2i3 3. −14i5 4. (−i)3
The opposition to current in an electrical circuit is called its impedance. The impedance z in a parallel circuit with two pathways satisfies the equationwhere z1 is the impedance (in ohms) of pathway 1 and z2 is the impedance (in ohms) of pathway 2. (a) The impedance of each pathway in a parallel
Cube each complex number. (a) −1 + √3i (b) −1 - √3i
1. The sum of two complex numbers is always a real number. 2. There is no complex number that is equal to its complex conjugate. 3. - i √6 is a solution of x4 − x2 + 14 = 56. Determine whether the statement is true or false. Justify your answer.
Find the missing values.What pattern do you see? Write a brief description of how you would find i raised to any positive integer power.
1. The coordinate system shown below is called the complex plane. In the complex plane, the point (a, b) corresponds to the complex number a + bi.Match each complex number with its corresponding point. (i) 3 (ii) 3i (iii) 4 + 2i (iv) 2 ˆ’ 2i (v) ˆ’3 + 3i (vi) ˆ’1
Prove that the complex conjugate of the product of two complex numbers a1 + b1i and a2 + b2i is the product of their complex conjugates.
Prove that the complex conjugate of the sum of two complex numbers a1 + b1i and a2 + b2i is the sum of their complex conjugates.
The temperature T (in degrees Fahrenheit) of saturated steam increases as pressure increases. This relationship can be approximated by the modelT = 75.82 ˆ’ 2.11x + 43.51 ˆšxwhere x is the absolute pressure (in pounds per square inch). (a) The temperature of saturated steam at
The total voting-age population P (in millions) in the United States from 1990 through 2014 can be modeled by P = 184.64 + 0.7524t2/1 + 0.0028t2, 0 ≤ t ≤ 24 where t represents the year, with t = 0 corresponding to 1990. (Source: U.S. Census Bureau)(a) In which year did the total voting-age
1. The demand for a video game can be modeled by p = 40 - âˆš0.01x + 1 where x is the number of units demanded per day and p is the price per unit. Find the demand when the price is $37.55.2. A power station is on one side of a river that is 34 mile wide, and a factory is 8 miles downstream on
Working alone, you can tile a floor in t hours. When you work with a friend, the time y (in hours) it takes to tile the floor satisfies the equation 1/t + 1/t + 3 = 1/yFind the time it takes you to tile the floor working alone when you and your friend can tile the floor in 2 hours working together.
Working alone, you can paint a fence in t hours. Working with a friend, the time y (in hours) it takes to paint the fence satisfies the equation1/t + 1/t + 2 = 1/yFind the time it takes you to paint the fence working alone when you and your friend can paint the fence in 3 hours working together.
1. The distance d a spring is stretched can be calculated using the formula d = √2U/k where U is the potential energy and k is the spring constant. Solve the formula for U. 2. The circumference C of an ellipse with major axis length 2a and minor axis length 2b can be approximated using the
The equation √x + 10 - √x − 10 = 0 has no solution. Determine whether the statement is true or false. Justify your answer.
Find x such that the distance between (3, −5) and (x, 7) is 13.
Find y such that the distance between (10, y) and (4, −3) is 10.
Describe the error(s).
The figure shows a glass cube partially filled with water.(a) What does the expression x2 (x - 3) represent? (b) Given x2 (x - 3) = 320, explain how to find the capacity of the cube.
Find a and b when the solution of the equation is x = 9. (There are many correct answers). Consider an equation of the form x + |x - a| = b, where a and b are constants
Write a short paragraph listing the steps required to solve this absolute value equation and explain why it is important to check your solutions.
Find a and b when the solution of the equation is x = 20. (There are many correct answers.) Consider an equation of the form x √x - a = b, where a and b are constants
Write a short paragraph listing the steps required to solve this radical equation and explain why it is important to check your solutions.
Solve the equation. Check your solutions. 1. x4 − 4x2 + 3 = 0 2. x4 − 13x2 + 36 = 0
Solve the equation, if possible. Check your solutions. 1. √5x - 10 = 0 2. 6 - 2√x = 0 3. √x + 8 − 5 = 0
Solve the equation. Check your solutions. 1. (x - 5)3/2 = 8 2. (x + 2)2/3 = 9 3. (x2 - 5)3/2 = 27
Solve the equation. Check your solutions. 1. 6x4 − 54x2 = 0 2. 36x3 − 100x = 0 3. 5x3 + 30x2 + 45x = 0
Solve the equation. Check your solutions. 1. x = 3/x + 1/2 2. 4/x - 5/3 = x/6 3. 1/x - 1/x + 1 = 3
Solve the equation. Check your solutions. 1. |2x - 5| = 11 2. |3x - 2| = 7 3. |x| = x2 + x - 24
(a) Use a graphing utility to graph the equation, (b) Use the graph to approximate any x-intercepts, (c) Set y = 0 and solve the resulting equation, (d) Compare the result of part(c) with the x-intercept(s) of the graph.1. y = x3 − 2x2 − 3x2. y = 2x4 − 15x3 + 18x2
Find the real solution(s) of the equation. (Round your answer(s) to three decimal places) 1. 3.2x4 − 1.5x2 = 2.1 2. 0.1x4 − 2.4x2 = 3.6 3. 7.08x6 + 4.15x3 = 9.6
Write an equation that has the given solutions. (There are many correct answers.) 1. -4, 7 2. 0, 2, 9 3. -7/3, 6/7
A college charters a bus for $1700 to take a group of students to see a Broadway production. When 6 more students join the trip, the cost per student decreases by $7.50. How many students were in the original group?
Three students plan to divide the rent for an apartment equally. When they add a fourth student, the cost per student decreases by $150-per month. What is the total monthly rent for the apartment?
An airline runs a commuter flight between Portland, Oregon, and Seattle, Washington, which are 145 miles apart. An increase of 40 miles per hour in the average speed of the plane decreases the travel time by 12 minutes. What initial average speed results in this decrease in travel time?
A family drives 1080 miles to a vacation lodge. On the return trip, it takes the family 212 hours longer, traveling at an average speed that is 6 miles per hour slower, to drive the same distance. Determine the average speed on the way to the lodge.
1. You deposit $2500 in a long-term investment fund in which the interest is compounded monthly. After 5 years, the balance is $2694.58. What is the annual interest rate? 2. You deposit $6000 in a long-term investment fund in which the interest is compounded quarterly. After 5 years, the balance is
1. An airline offers daily flights between Chicago and Denver. The total monthly cost C (in millions of dollars) of these flights can be modeled by C = √0.2x + 1 where x is the number of passengers (in thousands). The total cost of the flights for June is 2.5 million dollars. How many passengers
1. The set of all real numbers that are solutions of an inequality is the ________ ________ of the inequality. 2. The set of all points on the real number line that represents the solution set of an inequality is the ________ of the inequality. 3. It is sometimes possible to write two inequalities
1. The mean hourly wage W (in dollars) of chemists in the United States from 2000 through 2014 can be modeled by W = 0.903t + 26.08, 0 ≤ t ≤ 14 Where t represents the year, with t = 0 corresponding to 2000. (Source: U.S. Bureau of Labor Statistics) (a) According to the model, when was the mean
1. The times required to perform a task in a manufacturing process by approximately two-thirds of the workers in a study satisfy the inequalityˆ£t ˆ’ 15.6ˆ£ ‰¤ 1.9Where t is time in minutes. Determine the interval in which these times lie.2. A geographic
1. You stop at a self-service gas station to buy 15 gallons of 87-octane gasoline at $2.22 per gallon. The gas pump is accurate to within 1/10 gallon. How much might you be undercharged or overcharged?2. The side length of a square is 10.4-inches with a possible error of 1/16 inch. Determine the
Determine whether the statement is true or false. Justify your answer.If a, b, and c are real numbers, and a < b, then a + c < b + c.If a, b, and c are real numbers, and a ≤ b, then ac ≤ bc.If −10 ≤ x 8, then −10 ≥ x and -x ≥ -8.If −2 < x < −1, then 1 < -x < 2.
1. Give an example of an inequality whose solution set is (-ˆž,ˆž).2. The graph shows the relationship between volume and mass for aluminum bronze. (a) Estimate the mass when the volume is 2cubic centimeters. (b) Approximate the interval for the mass when the volume is greater
Solve the inequality. Then graph the solution set. 1. 4x < 12 2. 10x < −40 3. −2x > −3
Solve the inequality. Then graph the solution set. 1. 1 < 2x + 3 < 9 2. −9 ≤ −2x − 7 < 5 3. 0 < 3(x + 7) ≤ 20 4. −1 ≤ − (x − 4) < 7 5. -4 < 2x - 3 / 3
Solve the inequality. Then graph the solution set. (Some inequalities have no solution.) 1. ∣x∣ < 5 2. ∣x∣ ≥ 8 3. ∣x / 2∣ > 1 4. ∣x / 3∣ < 2 5. ∣x - 5∣ < - 1 6. ∣x - 7∣ < -5

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