Sunday, January 11, 2015

Word Problems

You live in a two story home (see attached page for floor plan, assume all walls are one foot thick, and each room is eight feet tall), you have decided to turn the house into an aquarium.

  1. How much sand will you need to buy to cover the floors to a depth of three inches?

  2. What will the weight of that sand be?  (Sand has a density of 80 to 100 pounds per cubic foot.  Assume it averages 90 pounds per cubic foot.)

  3. What would the sand weigh on the surface of the moon?

  4. Would it have been better to buy those colored little rock things that are in some fish tanks? (Show your work.)

  5. What will be the PSI exerted by the sand upon the floors?  (Give answers for the ground floor and upper floor separately and label both.)  Will the floors withstand the force?

  6. You will need to waterproof the house so that it does not leak.  You plan to do it with a revolutionary new type of tarp which is massless, volumeless, and completely water tight.  What square footage will you require?

  7. The tarp comes in the following sizes:
    8 ft x 20 ft for $20.97 - - - - 12 ft x 24 ft for $ 31.97 - - - 18 ft x 24 ft for $56.97
    24 ft x 24 ft for $75.97 - - - 30 ft x 50 ft for $198.00 - - - 40 ft x 40 ft for $211.00
    30 ft x 60 ft for $238.00

    Assuming that you need at least four inches of overlap to ensure the structure remains water tight, what combination of tarps will you buy to do the job with the lowest cost?  (Show your work.)

  8. How many gallons of water will be required to fill the house? (Show your work.)

  9. Jake from down the street thinks he can get you a better deal on the sand, the tarp, and the water, but his supplier requires that measurements be made in metric.  Convert the previous answers (1, 2, 3, 5, 6, 8) to metric.

  10. You want to replace the windows and doors with clear material so that you can look in on the fish.  Assume that water does not compress (this is usually a safe assumption as it does not compress much.)  The density of water is 62 pounds per cubic foot.  Recalling that in a liquid pressure at a given depth is exerted equally in all directions, how much pressure will doors need to withstand?  How much will windows on the ground floor?  How much will windows on the second floor?

  11. Would you be better off using glass or plastic as your transparent material? (Show your work.)

  12. What is the force exerted by the water and the sand on an average square foot of flooring
    a) on the ground floor?

    b) on the upper floor?

  13. Sea water has a density of 64 pounds per cubic foot.  What would the previous measurements (10, 12) be if you decided to make the house a salt water tank instead?

    (For questions 14-17 assume that you settled on fresh water.)
  14. You don't want the water to be stagnant so you plan to set up small water movers to develop a current in the house-tank.  You have ten such water motivators.  Where should they be placed?  (Show your work.)

  15. You don't want the current to be too small because you think that might lead to dead zones developing. instead you have desired to move water at the maximum velocity that will not tear the house apart.  What is that velocity?  (Hint: use Bernoulli's principle to help determine the effect doorways will have on this calculation.)

  16. Alice from up the street has convinced you that maximum current is a waste.  What is the minimum velocity your water motivators need to induce (using the placements from question 14) in order for 90% of the water in the house to circulate throughout the house in a 24 hour period?

  17. Given that the volume of water in the house (see your answer to question 8) is finite, only a certain number of fish will fit into it.  Larger fish require more free space than smaller ones.  The higher the proportion of large fish you buy, the fewer fish you will buy overall.  The higher the proportion of small fish you buy, the more fish you will buy overall.

    What is the most awesome ratio of small to large fish?  (Show your work.)

  18. You've decided to make the house have a mix of fresh and salt water zones.  How many times did the average student just say, "Fuck"?

  19. Using the water motivator positions determined in 14 and the water motivation velocity determined in 16, determine how long it would take for the water in the house to become 95% homogenized if you did nothing to maintain the different salinities.

  20. You want the house to be 40% fresh water, 40% salt water, and 20% brackish water (by volume) describe the system required to maintain this distribution (within 2% tolerance) in perpetuity.

  21. How many times will the average bull shark pass through different salinity zones in a year?  (Show your work.)

  22. The octopus has been eating your sharks.  If the octopus draws an average of 15 tourists primarily college students, and the sharks draw an average of 30 primarily children, is it most cost effective to get rid of the octopus, stop buying new sharks, or continuing to replace the sharks as they are eaten under the following conditions?

    a) children get a 40% discount but tend to bring 1.25 adults for every two children, college students get a 20% discount but tend to bring .5 non-student adults per every 10 students, and 80% of the adults accompanying children are non-students.

    b) Children get a 10% discount but tend to bring 1 non-student adult for every two children, college students get no discount.

    c) Children get no discount on weekends but get in free on week days.  College students get in free on Tuesdays but pay full price on other days.

    d) The octopus has requested you buy it larger sharks, these last few were kind of scrawny.

  23. A leak has developed in the floor and begun to flood your high school friend's secret lab where ze was trying to create a genetic combination of shark and octopus.

    If the equation for the size of the hole is X(t) = X(0) + k * dX(t) * P(t) where X is the size in square feet of the hole, X(0) is the size of the hole when it was first noticed, k is 3.14/(6ft), dX(t) is the change in the size of the hole at time t, and P(t) is the length of the perimeter of the hole in feet at time t (assume a circular hole), how long will it take before:

    a) The basement lab is half full?

    b) The computers in the basement lab get wet and short out?

    c) The specimens escape into the local sewer system?

    d) You have to plug the hole with a tourist to stop your fish from running out of water?

  24. You live in Iowa, the state with the most EF5 tornadoes per square mile.  After the house is destroyed by one, what are the odds (use p-value) that the SeeFee channel will pay you to be in the documentary, "Sharknado In Real Life"?

  25. What is the maximum speed a shark octopus hybrid can move on land, as measured in furlongs per fortnight?


Extra Credit:

  1. Which fish spent the most time in the den?
  2. Which fish spent the most time in the kitchen?
  3. Which fish spend the most time in the living room?
  4. What is a parametric equation for the movement of the octopus through the bedroom when it is stalking prey?


  1. 18. You've decided to make the house have a mix of fresh and salt water zones. How many times did the average student just say, "Fuck"?

    Question: do we need to take into account variation in length and volume of "Fuck"s, or are they all weighted equally?

    1. You win.

      I'm not totally sure what you win, but you definitely win something.

  2. I think you have flooded your house once too often....

  3. Please, please,please enter this for xkcd's What-If.


    1. Randall Munroe, physics student and webcartoonist, added a series of popular science essays to his repertoire called What If?. Many of the questions in this post would fit right in - I am reminded of "Cornstarch", for example.