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mathematics
calculus

Questions and Answers of Calculus

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
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
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
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
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 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
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
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
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.
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
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
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)
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
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
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
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
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 +
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
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
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
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
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
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)
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
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
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
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
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
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 =
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
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
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
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
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
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
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
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.
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
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
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
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
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. 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
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 ≤
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
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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