What is Word problem: Definition and 499 Discussions

In mathematics and computer science, a word problem for a set S with respect to a system of finite encodings of its elements is the algorithmic problem of deciding whether two given representatives represent the same element of the set. The problem is commonly encountered in abstract algebra, where given a presentation of an algebraic structure by generators and relators, the problem is to determine if two expressions represent the same element; a prototypical example is the word problem for groups. Less formally, the word problem in an algebra is: given a set of identities E, and two expressions x and y, is it possible to transform x into y using the identities in E as rewriting rules in both directions? While answering this question may not seem hard, the remarkable (and deep) result that emerges, in many important cases, is that the problem is undecidable.
Many, if not most all, undecidable problems in mathematics can be posed as word problems; see the list of undecidable problems for many examples.

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  1. D

    Word Problem with Constant 'e'

    Homework Statement For living organic material, the ratio of the number of radioactive carbon isotopes (carbon-14) to the number of non-radioactive carbon isotopes (carbon-12) is about 1 to 10^12. When organic material dies, its carbon 12 content remains fixed whereas its radioactive...
  2. M

    Help with an Algebra 2 word problem

    Homework Statement a glassblower can produce a set of simple gasses in about 2 hours. when the glassblower works with an apprentice, the job takes about 1.5 hours. How long would it take the apprentice to make a set of glasses when working alone? Homework Equations none The Attempt...
  3. Char. Limit

    Derivative Word Problem - Trig Needed?

    Homework Statement Now this is a problem that my sister had on her Calculus homework, and I can't seem to figure it out. I believe a similar triangles argument is necessary, but I'm not sure, and trig always was my weak spot. The problem is as follows: Homework Equations The Attempt at a...
  4. L

    Classic combined rates of work word Problem

    This is not a homework question. It is a question my sister asked me, and although I could solve it, I could not explain some parts of the solution. The question is: John and Mary are working on a job together. If John does it alone, it will take him 7 days, while Mary can do it alone in...
  5. K

    A confusing word problem. Can any one solve it?

    Homework Statement A 240gram toy car is driven by an electric motor that has a constant power output. The car can climb a 24° incline at 16m/s and can travel on a horizontal table at 39m/s. The frictional force retarding the motion is kv, where k is a constant and v is its speed. How steep...
  6. D

    How to Calculate Population Change in the United States?

    Homework Statement Let b(t) be the number of births in the United States at the end of year t, and let d(t) represent the number of deaths in the United States at the end of year t. If p(t) is the population of the United States at the beginning of year t, write a function c(t) that...
  7. P

    DiffyQ Word problem (growth rate)

    I have been trying to figure this problem out for more then 90 minutes and I think I'm past the point where I'm being productive: A population P satisfies the differential equation: P^'(t)=10^{-5}P(t)(15000-P(t)) For what value P(0) of the initial population is the initial growth rate P'(0)...
  8. M

    Word problem, interested in the reasoning process

    Homework Statement A 6-gallon radiator is filled with a 40% solution of antifreeze in water. How much of the solution must be drained and replaced with pure antifreeze to obtain a 65% solution? The answer is given: Ans. 2.5 gallons The Attempt at a Solution Let x be the amount of added...
  9. N

    Solving Word Problems: Velocity and Acceleration in s(t)=t2+tlnt Function

    The function s(t)=t2+tlnt gives the position from time [1,e] find: a. the velocity at time t b. the acceleration at time t c. describe the motion of the object: when is it moving forward or moving backward. also compute the total distance traveled during the interval d. when is the object...
  10. M

    General question about ang momentum throug word problem

    a person with mass m stands at the outer edge of a round platform whose radius is r and whose mass is m2. the platform is rotating at a speed of ...rad/s. Assuming the platform to be a free rotating disk, what would be its angular speed if the person walked to its center? Then the answer book...
  11. C

    Tsunami word problem (waves practice)

    Homework Statement A Tsunami has been generated as a plane wave 4000km off the shore of California. The sea surface displacement is found to be 3 meters and the wavelength of the tsunami is 100km. You can assume a depth of 4000m from the location of wave generation to shore. a.) Calculate...
  12. M

    Newton 2nd Law word problem help

    "Newton 2nd Law" word problem help Homework Statement When the click beetle jumps in the air, its acceleration upward can be as large as 400.0 times the acceleration due to gravity. (An acceleration this large would instantly kill any human being.) For a beetle whose mass is 40.00 mg...
  13. P

    Tree Cutting Word Problem: Why Does the Short Tree Hit the Ground First?

    Homework Statement a tall tree and a short tree of the same mass density and width are cut at the base at the same time. why does the small tree hit the ground first? Homework Equations none. its a word problem The Attempt at a Solution gravity id pushing in different ways?
  14. P

    Teeter Totter Balance: Why Does the Lighter Person's End Go Down?

    Homework Statement suppose that a heavy person and a light person are balanced on a teeter totter made of plank wood. each person moves toward the fulcrum 25cm. why will the lighter persons end go down?? Homework Equations not needed The Attempt at a Solution something about forces...
  15. M

    What is the Initial Velocity of the Racket in a Momentum Word Problem?

    momentum word problem... Problem: A tennis ball initially moves with a velocity of 30.0 m/s horizontally to the right. A tennis racket strikes the ball, giving it a velocity of 30.0 m/s to the left. Treat the collision as elastic and the ball and racket as particles of mass 0.0500= m1 kg...
  16. M

    Optimized Solving a Physics Word Problem - Can You Help?

    I added two pictures.. One of the word problem and one of my diagram I made.. My work: 1.Wnet = Kf - Ki -Ugf = Kf... -mgy = mv^2/2 ... - gy = v^2/2 2. x^2 + y^2 = 1780^2 dy/dx = -x/y = velocity (v) at point (x,y) 3. gy = x^2/(2y^2) 4. gy = (1780^2 - y^2)/(2y^2) ...
  17. M

    Calculating Work Done by Tension in Vine for Tarzan

    Tarzan, who weighs 875 N, swings from a vine through the jungle. How much work is done by the tension in the vine as he drops through a vertical distance of 4.00 m? The answer is zero. Why is this?
  18. G

    Max Profit Calc: Sell Farmer's Crop by Oct 10

    Homework Statement A farmer can sell his crop for p= -t^2+60t+100 dollars per ton, where p is price and t is the number of days after Oct. 1st. Additionally, the farmer must sell his crop on or before Oct. 10 to pay his outstanding credit card bills. On what day should he sell to maximize...
  19. P

    Finding Formulas for c, μx, and Var(X) in a Center of Mass Word Problem

    Let p be a positive constant. Suppose that a random variable X has probability function f(x)=cxp(1-x) for 0≤x≤1. Find formulas for c, μx, and Var(X) in terms of p. I know Var(X)=∫(x-μx)2f(x)dx I know that I did not show any work so please just give me a hint (I'm not asking you to solve...
  20. M

    Partial Derivatives used in a manufacturing scenario word problem

    Homework Statement A) if http://www4d.wolframalpha.com/Calculate/MSP/MSP10819hf7feh5hf5hh7e00004ga728h1d42i6h0c?MSPStoreType=image/gif&s=18&w=161&h=20 Find the partial derivatives? B) Suppose you are manufacturing blackboards and whiteboards and that P is your monthly profit when your...
  21. A

    How Many Items Must Be Sold for a Company to Break Even?

    Homework Statement A company sells a certain product and determines that the best price that each item/product must be sold for is given by the formula P=1,620 Square Root X Over X. AKA P=1,620 Square Root X divided by X. X is the # of items sold. Each item cost's $40 to produce4. All items...
  22. P

    What is the Relationship Between Net Force, Speed, and Time for a Moving Object?

    Homework Statement An object is moving to the right, and experiencing a net force that is directed to the right, The magnitude of the force is decreasing with time. the speed of the object is: constant or increasing?? Homework Equations none? The Attempt at a Solution constant??
  23. R

    Microsoft Word Problem - Help Fix Red Lines & Equation Issues

    Ok I know this isn't exactly a direct physics problem but I wasn't sure where to post this. Im writing up a lab report in word 2008 mac and whenever I insert an equation it pushes the text down a little and it sits above the text even though its all technically still the same line. Also the...
  24. A

    Polynomial Functions (word problem)

    Homework Statement A quartic function has x-intercepts of -5, -1, 2, and 4. The point (3.-8) lies on the curve. Determine an equation of the function. Homework Equations The Attempt at a Solution y = k (x+5)(x+1)(x-2)(x-4) Expanded form: f(x)=kx^4 + 9x^2 - 38x + 40 <---My...
  25. D

    Bayes Probability HIV Word Problem

    In this problem, assume researchers have constructed a risk score for HIV for the U.S. population, which is a function of risk factors such as frequency of unprotected sex, use of intravenous drugs, having another sexually trans- mitted infection, etc. Assume each risk factor measured is...
  26. N

    Word problem Polynomialf(x)

    Homework Statement A rectangular shipping container that the Food Bank uses to store their tinned food, has a volume of 2500 cm3. The container is 4 times as wide as it is deep, and 5cm taller than it is wide. What are the dimensions of the container Homework Equations The...
  27. N

    Word problem polynomial f(x) help

    Homework Statement 2 People build a sandcastle of 36cm by 48cm by 60cm. Later they learn that the box will be smaller; a certain amount will be taken off each of the length width and height Homework Equations The Attempt at a Solution a) write a p(x) > (36-x) (48-x) (60-x)...
  28. N

    Polynomial f(x) word problem help

    Homework Statement Kaylas lemonade stand has experienced rises /falls in sales caused by temperature changes over 2 summers. Her sales over the first two weeks of summer were tracked according to the model S(x) = x^3 -12x^2 +36x, where x is the number of days and S(x) is the number of sales...
  29. M

    ODEs: Word problem involving (I think) phase lines

    Homework Statement If the population y of rats on a farm at time t (in weeks) satisfies: dy/dt = -y(y-100)/50 then how many rats per week should be killed to eradicate the population? Homework Equations None known. The Attempt at a Solution The ODE dy/dt is autonomous, so I can...
  30. M

    Solving Word Problem - Picture Guide

    I added a picture my work is inside the picture...
  31. J

    Vectors and angles word problem

    A snow-covered ski slope makes an angle of θslope = 29.0° with the horizontal. When a ski jumper plummets onto the hill, a parcel of splashed snow is thrown up to a maximum displacement of 1.70 m at θsnow = 19.0° from the vertical in the uphill direction as shown in the figure below. (a)...
  32. J

    How Do You Calculate the Resultant Displacement in a Football Pass Scenario?

    In a game of American football, a quarterback takes the ball from the line of scrimmage, runs backward for 10.5 yards, and then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a forward pass 55.0 yards straight downfield perpendicular to the line of...
  33. N

    How to draw a circuit from a word problem.

    Homework Question: Four identical 4Ω resistors have been connected in such a way that their total resistance is 4Ω. Draw a diagram to show how they are connected. My Question: How do you solve a question like this? How do you classify it as a parallel or series circuit or does it even...
  34. H

    Maximize Revenue from ticket sales word problem

    Homework Statement Homework Statement [/b] A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket prices set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average...
  35. N

    What is the change in horizontal distance from low tide to high tide?

    Homework Statement A cargo ship is tied up at the dock. At low tide, a 12-m long unloading ramp slopes down from the ship to the dock and makes an angle of 30 degrees to the horizontal. At high tide, the ship is closer to the dock, and the unloading ramp makes an angle of 45 degrees t othe...
  36. nukeman

    Arithmetric Sequence word problem. Thanks

    Homework Statement Ok here is the problem: 100 loaves of bread are to be distributed such that the quantities form an arithmetric sequence, among 5 people. The first 2 together receive 1/7 of the remaining 3, together. How much (loaves and fractions) does each receive? Homework...
  37. N

    What is the Inverse Function of a Dropped Object's Height Over Time?

    Homework Statement If an object is dropped from a height of 80m , its approximate height, h(t) metres above the ground t seconds after being dropped is given by the function h(t) = -5t^2 + 80 a) graph the functions b) find and graph the inverse c) is the inverse a function d) what does...
  38. N

    Word problem functions

    Homework Statement Sloping roof : The diagram shows a set of coordinate axes superimposed on the cross section of a sloping roof of height h metres and width w metres. The equation of one half of the cross section is y=-0.7x + 1.9 a) What is the equation of the other half of the...
  39. K

    Help with an investment matrix word problem

    Help with an investment matrix word problem! Can anyone give me the 3 equations in general form that I need to solve this matrix? A person will put 30% of his investment into a stock fund that earns a 9% annual rate of return. He will put 20% in precious metals currently growing at 40% and...
  40. N

    What is the solution to this parabola word problem?

    Hi all. I'm a nurse in the middle of a nightmare math class. It is second half of college algebra but it may as well be Greek. The professor is a PhD in engineering and is teaching way over my head- and others from what I hear. I have never failed a class in my life but this may be a first...
  41. C

    Word Problem: Finding a Number with -5 and 2 in the Equation

    The word problem: If the product of a number and -5 is reduced by 2, the result is 26 less than twice the opposite of the number. What is the number? I wrote it out like this: n x (-5) - 2 = 2(-n) - 26 I'm really at a loss here. Normally when there's a variable, my teacher says to get the...
  42. J

    Optimizing Area in a Semi-Circle

    1. Find the dimensions of the rectangle with the largest area that can be inscribed in the upper semi-circle given by x^2+y^2 ≤ 16, y≥0. 2. I thought I'd use A=lw 3. This is but a guess..so take it with a grain of salt.. height=2x base= x^2+y^2 A(x) = 2x(x^2+y^2) = 2x^3+2xy^2 A'(x) =...
  43. J

    Hyperbola Word Problem: Meteorologists and the Speed of Sound

    Two amateur meteorologist, living 4km apart (4000m), see a storm approaching. The one farthest from the storm hears a loud clap of thunder 9 sec after the one nearest. Assuming the speed of sound is 340m/sec, determine an equation that models possible locations for the storm at that time...
  44. S

    Come up with a sine equation for the following word problem.

    niagra falls skywheel, a ferris wheel has a diameter of 50.5 m and rotates 1 revolution in 2 minutes. the lowest point on the wheel is 2.5 m from the ground. This is the equation i have come up with, please tell me if this is right or not, and if it is not please come up with your own...
  45. QuarkCharmer

    Calculating Percentage of Stannous Fluoride in Toothpaste Sample

    Homework Statement A 10.000g sample of toothpaste containing stannous fluoride gives a 0.145g precipitate of stannous phosphate. What is the percentage of SnF_2 in the toothpaste sample?Homework Equations (Actual yield)/(Theoretical yield) * 100% 3SnF_{2}_{(aq)} + 2K_{3}PO_{4}_{(aq)}...
  46. W

    Solving Matrix Word Problem: Amanda, Bryce & Corey

    Amanda, Bryce, and Corey enter a race in which they have to run, swim, and cycle over a marked course. Their average speeds are given in the table. Corey finishes first with a total time of 1 hours 45minutes, Amanda comes in second with a time of 2 hours 30 minutes, and Bryce finishes last with...
  47. G

    Find a linear system of equations from the word problem?

    The problem: "A commercial customer orders 81 gallons of paint that contains equal amounts of red paint, green paint, and blue paint - and hence, could be prepared by mixing 27 gallons of each. However, the store wishes to prepare this order by mixing three types of paint that are already...
  48. M

    Solving a Trig Word Problem Involving Pipes

    Homework Statement Six Identical pipes, each with radius of 1 foot are ties tightly together with a metal band... Find the length of the metal band... I posted a picture... My strategy is dividing the figure in half Then finding the measure of the two arcs in red ( which should be...
  49. M

    Finding the Radius (word problem)

    Homework Statement If a pipe is between two blocks... one block being 1 inch tall and the other block being 2 inches tall. Between the two blocks is an arc that is 6 inches... What is the radius? Can some one please help me... I have no Idea what to do.. thank you.
  50. A

    Modeling Technology Adoption in a Community: Solving a Differential Equation

    A new technology is introduced into a community of 5000 individuals. If the rate dN/dx at which the technology spreads through the community is JOINTLY PROPORTIONAL to the number of people who use the technology AND the number of people who do not use it, (1)WRITE A DIFFERENTIAL EQUATION FOR...
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