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. K

    Standard Deviation Word Problem

    Homework Statement It has been projected that the average and standard deviation of the amount of time spent online using the Internet are, respectively, 14 and 17 hours per person per year (approximately normally distributed). What value is exactly 1 standard deviation below the mean?Homework...
  2. F

    Why Does Running in the Rain Get You More Wet?

    If you stand motionless under an umbrella in a rainstorm where the drops fall vertically, you remain relatively dry. However, if you start running, the rain begins to hit your legs even if they remain under the umbrella. Why?Now I know the answer, it's because when you run horizontally, your...
  3. E

    D.E. Word Problem (Extruded Volume, constant decrease in volume)

    This was the problem before the one I am asking about (backstory maybe?) - Maybe relevant to my current problem \frac{dh}{dt}=-\frac{A_{h}}{A_{w}}\sqrt{2gh} Problem statement: It seems that, by placing marks on the tank corresponding to the water level at intervals of, say, one hour one...
  4. S

    Word problem: Asking to solve for average velocity and average speed

    [b]1. 1. To get to a concert in time, a harpsichordist has to drive 124 miles in 2.01 hours. (a) If he drove at an average speed of 53.0 mi/h in a due west direction for the first 1.18 h, what must be his average speed if he is heading 30.0° south of west for the remaining 49.8 min? (b)...
  5. S

    Magnitude and Direction Word Problem

    [b]1. The pilot of a small plane finds that the airport where he intended to land is fogged in. He flies 45 miles west to another airport to find that conditions there are too icy for him to land. He flies 20 miles at 15° east of south and is finally able to land at the third airport. (a)...
  6. S

    Calculating Final Velocity with Acceleration: Space Shuttle Example

    Homework Statement The space shuttle undergoes an acceleration of 53.2 meters/second^2. How fast is it traveling at the end of 55.2 seconds? Homework Equations not sure? The Attempt at a Solution All i really need is the set up. I've gone through a page of equations in my...
  7. I

    Word Problem Beginning Calculus

    1. Homework Statement [/b] A cyclist is riding on a path modeled by f(x)= 0.08x where f and f(x) are measured in miles. Find the rate of change in elevation when x = 2 Homework Equations (f(∆x + x ) - f(x))/∆x ∆x\stackrel{lim}{\rightarrow} 0 The Attempt at a Solution I...
  8. Williams.235

    Trig Word Problem: Finding Height of a Baloon

    Homework Statement While taking a ride in a hot-air balloon, Fran wonders how high he is. To find out, he chooses a landmark that is to the east of the balloon and measures the angle of depression to be 54 degrees. A few minuets later, after traveling 100 feet east, the angle of depression...
  9. S

    Introductory kinematics word problem

    Homework Statement A competitor is aiming to complete a 1500m wheel-chair race in less than 4.0 minutes. After moving at a constant speed for exactly 3.5 minutes, there were still 240m to go. What must his acceleration before the remaining distance if he were to finish the race on time...
  10. M

    Area Word Problem + Graph Problem

    PROBLEM #24- A goat rancher is going to fence off a rectangular pen in the middle of a large open area. Build an equation to help determine the maximum area (to the nearest sq ft) that can be fenced off if 1200 ft of fencing is avilable. Include a diagram as a part of your solution.) I have...
  11. W

    Calculating Insurance Premiums with Variable Damages and Probabilities

    Homework Statement Suppose that one year, an insurance company incurred dollar damages, X, in four different amounts with probabilities, p(x), shown below: X 0 1000 5000 10000 p(x) 0.7 0.2 0.08 0.02 If the company offers a $500 deductable and wants and wants an...
  12. W

    How Is the Probability Mass Function Calculated for Identifying Blood Type A+?

    Homework Statement There is a group of four blood donors: A, B, C and D. Only A has type A+ blood. Four blood samples, one from each individual, will be typed in random order until an A+ individual is identified. Let Y = {number of typings necessary to identify an A+ individual}. Compute...
  13. N

    What is the optimal lower limit for solving an integral word problem?

    Homework Statement Here is the problem: The Attempt at a Solution I found the intersection point to be (1.10,2.78). Now I don't know what to use for the lower limit.
  14. N

    Integral Word Problem EXACT PROBLEM Incluced

    Homework Statement Here is the problem: The Attempt at a Solution I can see that as x increases, the deeper the water gets, but I can't figure out my limits of integration. If somecan just give me the limits, I think I would be all set.
  15. N

    Definite Integral Word Problem

    Homework Statement A rectangular pool is 10 meters wide and 25 meters long. The depth of water "x" meters from the shallow end of the pool is Part A. Write a left-hand Riemann Sum with 5 terms that approximates the volume of the water in the pool. Is your approximation an underestimate or an...
  16. T

    Solve Energy Word Problem: Find Height Swinging Rope w/ 3.5 m/s Vel

    Homework Statement A student is running at a speed of 3.5 m/s and grabs a long rope that is hanging vertically from the ceiling. How high can the student swing? Homework Equations For what we're learning right now, we should be using: PEi + KEi = PEf + KEf The Attempt at a...
  17. J

    How Deep Will Water Be in a Tank After Two Hours?

    Homework Statement A water tank is a rectangular box with a base which is square of side length 5 meters. At t=0 hours it has water to a depth of 2 meters. The rate water flows into the tank after t hours is 1+(t/2) m^3/hr. Make a graph of this on graph paper. Use your graph to find the depth...
  18. D

    Searchlight Rotation Rate in Related Rates Word Problem

    Homework Statement A man walks along a straight path at 5km/hr. A search light is located on the ground 30 metres form the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 20 metres from the point on the path closest to the searchlight...
  19. G

    Projectile Motion Word Problem

    Homework Statement From the roof of tower 1, a theif throws a money bag to an accomplice on the roof of tower 2, which is just west of tower 1. The two towers are separated by a mall. The defense attorney contends that in order to reach the roof of tower 2, the defendant would have had to...
  20. G

    Physics Word Problem; Projectile Motion

    Homework Statement Suppose a catcher is crouched down behind the plate wen he observes the runner breaking for second. After he gets the ball from the pitcher, he throws as hard as necessary to second base without standing up. If the catcher throws the ball at an angle of 30 degrees from the...
  21. H

    An exponential growth/decay word problem. If spraying of grasshoppers

    Homework Statement If spraying of grasshoppers with 800kg/km^2 of a certain insecticide will kill 40% of these insects, how much insecticide is needed to kill 98% of the grasshoppers? (assume exponential relationship) Homework Equations \intdP/dt=\intkp lnP=kt+c e^(kt+c)=PThe Attempt at a...
  22. S

    What Is the Equation of a Parabola for a Bridge Needing Specific Dimensions?

    Homework Statement 1. It is a word problem I need done by tomorrow morning. so please help: A new bridge is being constructed. The space between the support needs to be 1050 feet; the height at the center of the arch needs to be 350 feet. An empty tanker needs a 280 foot clearance to pass...
  23. S

    Word problem find equations then solve

    Homework Statement A certain amount of money saved for 1 year at a certain annual interest rate(simple interest) yielded 19.36$ in interest. if we had added $66 to the principal and reduced the interest rate by 3 percentage point(s), the interest would have been the same, Find the principal...
  24. B

    Solve Word Problem: Find a and b in T(t)=250-ae^(-bt)

    Homework Statement A yam is pust in an oven maintained at a constant temp of 250degrees C. Suppose that after 30 min the temp of the yam is 150degrees C and is increasing at a rate of 3degrees C/min. If the temp of the yam t minutes after it is put in the oven is modeled by...
  25. O

    Ship traveling cal word problem i keep getting the wrong answer

    At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.) Note: Draw yourself a diagram...
  26. A

    How to Solve Trigonometric Word Problems Correctly?

    Homework Statement http://img135.imageshack.us/img135/4150/trighelp.jpg Homework Equations The Attempt at a Solution http://img18.imageshack.us/img18/8480/workz.jpg Please help I'm totally confused, if anyone can help I can offer a reward of monetary value :).
  27. 6

    How to transform this word problem into a binomial distribution equation

    Homework Statement Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently a) What is the probability...
  28. T

    Quadratic Function Word Problem

    Homework Statement A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum...
  29. B

    Solving a Systems of Linear Equations and Matrices Problem

    Problem: To get the necessary funds for a planned expansion, a small company took out three loans totaling $25,000. Company owners were able to get interest rates of 8%, 9%, and 10%. They borrowed $1000 more at 9% than they borrowed at 10%. The total annual interest on the loans was $2190...
  30. T

    Interesting word problem posed

    So I had a student today ask me if I could come up with a simple 2 dimensional area problem. I was like -sure what is it. The question was: If a cow were strapped to a tether outside a 10' by 10' square barn, at the corner of a small outlet door, and had a 100' leash/tether, what would the...
  31. M

    Transformation matrix, vector algebra word problem

    Hi everyone. I am not sure if this problem belongs under the "Linear & Abstract algebra" section but it seemed like it may. Please let me know if there is a different section that would better fit this problem. So here is a word problem that is proposed: A solar panel is capable of rotating...
  32. H

    Proving Symmetry of A + A(Transpose) for Any Square Matrix A

    Homework Statement Show that A + A(transpose) is symmetric for any square matrix A. Homework Equations A symmetric matrice is equal to A=A(transpose) The Attempt at a Solution I said A + A(transpose) = A(transpose) + A(transpose) = (A + A) (transpose) = A + A I don't know...
  33. S

    Therefore, Bill and Gina would have the same number of coins, 80 each.

    Homework Statement The number of coins in Bill's collection is 80 less than twice the number in Gina's collection. Who has more coins? The answer is "Not enough information to answer the question." The Attempt at a Solution The way you do these problems is figure out the equation...
  34. J

    What is the Physics Behind a Rock Thrown on the Moon?

    a rock is thrown vertically upward fom the surface of the moon at a velocity of 24m/s(about 86km/h)reaches a height of s=24t-0.8t^2meters in t seconds. (a)find the rocks velocity and acceleration as functions of time?(accleration of gravity on the moon) (b)how long did it take the rock to...
  35. K

    Solve Tricky Word Problem: Motorcycle Cop & Red Ferrari 308 GTS

    Homework Statement A motorcycle cop, parked on the side of a highway is passed by a woman in a red Ferrari 308 GTS doing 90km/h. After a few attempts to get his cycle started, the officer roars off 2.00s later. At what average rate must he accelerate if 110km/hr is the top speed and he is to...
  36. M

    Word problem using derivatives - struggling with it

    Homework Statement the equation is PV = c; p = pressure, v = volume, c=constant. (also known as Boyle's law) the question: if volume is decreasing at a rate of 10cm^3/minute, how fast is the pressure increasing when the pressure is 100g/cm^2 and volume is 20 cm^3 Homework Equations...
  37. M

    Momentum (airplane landing, word problem)

    Homework Statement A 1.0 x 10^3 kg plane is trying to make a forced landing on the deck of a 2.0 x 10^3 kg barge at rest on the surface of a calm sea. The only frictional force to consider is between the plane's wheels and the deck; this braking force is constant and is equal to one-quarter of...
  38. A

    How Do You Calculate Instantaneous Rate of Change in an Electrical Circuit?

    Hi everyone, i am new to this website and i would like to ask one question that i don't quite get. we have started a new topic in class about rates of change Homework Statement An Electrical current in a cicruit varies with time according to C=(3s^3-s^2+5s)/(S^3+10) where currenct is...
  39. P

    Derivative word problem. Tricky

    Homework Statement A spaceship moves along y = x3 - 8x in the positive x-direction. Shutting off the engines at P allows it to move off along the tangent at P. Find P so that the ship can reach the point (4,0). Homework Equations Dunno The Attempt at a Solution So I start with...
  40. Z

    Absolute Minimum and Maximum Word Problem

    Homework Statement Find a number in the closed interval [1/2, 3/2] such that the sum of the number and its reciprocal is (a)as small as possible (b) as large as possible I am given the answer in the back of the book The answer to a is 1 The answer to be is 1/2 Homework...
  41. G

    What is the maximum product of two numbers when their sum is 100?

    Homework Statement Find two numbers whose product is a maximum if the sum of the first number and twice the second is 100 Homework Equations The Attempt at a Solution Alright so I think I might have the right answer but something just doesn't seem right. so first I named my...
  42. Y

    Solve Simple Word Problem: 100kg w/15% Moisture to 5% Dry Weight Fibers

    Homework Statement When paper is recycled, water is added to break the paper up into its original wood fibers. If 100 kg of paper with a moisture content of 15% is to be recycled, how much water (kg) must be added to the recycled paper to create a solution that is 5% by dry weight fibers...
  43. W

    Solve Basic Word Problem: Find Coins & Combos for $3.00

    Homework Statement I have nickels, dimes and quarters in a jar. There are 20 coins altogether and exactly twice as many dimes as nickels. The total value of the coins is $3.00 Find the number of coins of each type. Then find all possible combinations of 20 coins (nickels, dimes and...
  44. M

    Callable bonds & Yield to call date word problem

    Homework Statement A corporation sold a 20-year, 10% semi-annual coupon bond issue 2 years ago. The bonds are callable at 108 ( percent of face value) 5 years after issue and at 104 (percent of face value) 10 years after issue. If the bonds are currently priced at 102.5 (percent of face...
  45. M

    Compond interest word problem

    Homework Statement If the total interest earned on an investment at 8.2% compounded semiannually for 8.5 years was $1,175.98, what was the original investment? Homework Equations A=P(1+i)^n The Attempt at a Solution
  46. B

    One-Dimensional Motion Word Problem

    Homework Statement A stone is dropped from a height (h) and falls the last half of its distance in 4 seconds. (a) What is the total time of the fall? (b) From what height was the stone dropped? Homework Equations h=1/2(g)(T)² The Attempt at a Solution T=√2(T-4)
  47. M

    How much money did the soda machines generate during the second semester?

    Homework Statement In a West Texas school district the school year began on August 1 and lasted until May 31. On August 1 a Soft Drink company in- stalled soda machines in the school cafeteria. It found that after t months the machines generated income at a rate of f(t) = 500t/ (5t^2 +...
  48. M

    Physics word problem- velocity, acceleration, time

    Homework Statement two cars are traveling along a straight road, one behind the other. the first is traveling at a constant velocity of 12 m/s. the second, approaching from the rear, is traveling at 25 m/s. when the second car is 200m behind the first, the driver applies the brakes...
  49. B

    How Do You Calculate Snow Projection Components on a Slope?

    Homework Statement a snow-covered ski slope makes an angle of 35 degrees with the horizontal. when a ski jumper plummets onto the hill, a parcel of splashed snow projects to a maximum position of 5 m at 20 degrees from the vertical in the uphill direction. find the components of its maximum...
  50. B

    How Do You Calculate Displacement from Dallas to Chicago Using Vectors?

    Homework Statement a map suggests that atlanta is 730 miles in a direction of 5.00 degrees north of east from dallas. the same map shows that chicago is 560 miles in a direction of 21 degrees west of north from atlanta. modeling the Earth as flat, use this info to find the displacement from...
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