INTERESTING MATH PROBLEMS TO MAKE YOU THINK
Many students are afraid of math, and that is unfortunate. It is a powerful tool which is used every day to help us understand the universe around us. I like thinking about interesting problems that do not require very sophisticated math skills but which require you to think carefully and to work systematically toward an equally interesting solution. They are good for honing your thinking skills, your basic research skills, your mathematical skills, your logical skills, and your ability to organize and manipulate information without getting lost in the numbers. They also challenge you to develop a sense of what a reasonable answer might be. I often try to devise more than one approach to the solution. If two or more approaches yield the same solution, your confidence in the solution is increased. These are very useful, practical skills that
can make you a better thinker and student.
The problems below can only be solved through a careful, logical, multi-step process. For some of them you will need to know how to do scientific notation (because of the very large or small numbers involved). Your chemistry or physics textbook will explain the process, or go to this link: http://www.nyu.edu/pages/mathmol/textbook/scinot.html
I encourage you to try some of the problems below before looking at the solutions. You'll learn a lot about your thinking processes even if you fail to get the right solution on the first try.
I will add new problems and solutions as I find time.
1. THE PENNY PROBLEM
I'll start with a really easy one.
How many pennies, stacked one on top of another, would it take to reach a height of one mile?
Part of the challenge of these problems is figuring out what you need to know, doing some basic research when necessary (use Wikipedia), and proceeding from there.
2. THE QUARTER PROBLEM
You have $1.5 million dollars in quarters, and you want to stack them up diameter-to-diameter in a space that is 3 feet square. If all the stacks rise to more or less the same height, how tall will the stacks approximately be in feet?
(This should be obvious, but only work with whole not fractional quarters.)
3. THE POOL PROBLEM
A swimming pool is 50 meters long, 25 meters wide and 2.5 meters deep. It is 85% filled with water. How much does the water in the pool weigh in standard tons?
4. THE BB PROBLEM
Here's a problem for you chemistry students who deal often with moles in making calculations. It is easy to overlook the fact that a mole of anything (copper atoms, cupric sulfate molecules....or copper BBs) is an astronomically large number. This problem will give you a sense of how large a mole is.
You have a mole of BB's and want to spread them evenly across the length and breadth of the continental United States. How deep will they pile up?
(Assume that the BB's are stacked up diameter to diameter and that every other row is NOT nested between the BBs in the previous row. Choose the most appropriate English units to measure the depth of the pile--one that will give you the lowest possible number to express the depth.)
5. THE CONCRETE WEIGHT PROBLEM
This problem derives from an actual situation I had. I needed a 25 pound weight to keep proper tension on one of my long-wire amateur radio antennas. After solving the problem below, I cut the pipe, poured the concrete, and let it set up. When I weighed it later, it weighed almost exactly 25 pounds.
A standard bag of Fast-Setting Quikrete, when mixed with water according to directions on the bag, produces 0.45 cubic feet of concrete weighing 65 pounds. You want to pour the wet concrete mix into a length of plastic pipe with an inside diameter of 4" to create a weight weighing 25 pounds. To what length (expressed in feet and inches) should you cut the plastic pipe before filling it with wet concrete?
6. THE $100 BILL PROBLEM
The National Debt on August 23, 2010, stood at 13. 374 trillion dollars (see http://www.brillig.com/debt_clock/). This is the amount of money owed by the U.S. federal government to its various creditors. The national debt has more than doubled over the course of the last decade. It is difficult to imagine this magnitude of debt or in fact trillions of anything. This problem will give you a sense of how large 13.374 trillion is.
Suppose you could create a stack of $100 bills (the largest denomination in general use) until it amounted to $13.374 trillion dollars. How high would be stack be?
(Express your answer in the most appropriate English units. Assume a $100 bill is 0.01" thick.)
7. THE GIANT SEQUOIA PROBLEM
This problem was inspired by some comments recently made by a speaker in Chapel.
Approximately how many wooden pencils could be milled from the trunk of a giant sequoia tree 300 feet long and 35 feet in diameter?
8. THE NICKEL PROBLEM
This problem came up in a U.S. History class discussion about the intrinsic value of coins. While gold and silver have intrinsic value as precious metals, what about coins in common circulation in America today? Does the U.S. Nickel, for example, have intrinsic value or not?
The 24-hour peak commodity price for nickel on September 24, 2010, was $10.3464 per pound. The peak price for copper was $3.5885 per pound. Determine whether the U.S. Nickel had intrinsic value or not and determine its actual worth in fractional U.S. dollars on that date.
9. VOYAGER II AVERAGE VELOCITY PROBLEM
Voyager II was launched from Cape Canaveral, Florida, at 10:29:44 AM EST on August 20, 1977, and is still functioning after 35 years. It reached its calculated rendezvous point around Jupiter at 06:29:00 PM EST on July 9, 1979, when it was 5.05 AUs from the earth. It was launched at an unusually high velocity to allow it to escape earth's gravitational influence and take a quick, direct route to the Jupiter system.
Calculate Voyager II's mean (average) velocity between launch and rendezvous (its closest approach to Jupiter). Express your answer in miles per hour.
10. VOYAGER II DISTANCE PROBLEM
Voyager II's first photographs of Uranus were beamed back to earth at 12:59:00 PM EST on January 24, 1986. The radio signal carrying the photographic data then began a 2.75 hour trip toward earth. Meanwhile, Voyager II continued to travel away from earth at an average velocity of 28,400 miles per hour.
What was the distance between earth and Voyager II when the first photograph was received on earth? Express your answer in miles and in AUs.
11. VOYAGER II TRANSMITTER OUTPUT PROBLEM
Voyager II's X-Band transmitter had an RF output of only about 21.3 watts. That's an incredibly weak signal for reception across the 4 billion mile radius of the solar system and required the networking of large parabolic antennas across the world. The strength of the signal when received on earth is expressed in terms of Power Flux Density (PFD) in watts per square meter.
PFD = PG / (4 x pi x r x r)
P = 21.3 watts (transmitter RF output)
G = 65,000 (antenna gain)
r = distance from earth in meters (convert AUs below to miles and then to meters)
Calculate the PFD of Voyager II's X-Band transmitter when received on earth from Jupiter (5.2 AU), Saturn (9.6 AU), Uranus (19.2 AU), and Neptune (30.1 AU). Show that these power output levels decline with distance in accord with the inverse square law.
12. ANOTHER VOYAGER II TRANSMITTER PROBLEM
Voyager II will continue virtually forever into space at its present velocity unless it impacts something along the way, in accord with Isaac Newton's First Law of Motion (the Law of Inertia). Given the emptiness of space, the chance of a collision is relatively remote.
If Voyager II's X-Band transmitter were still functional after a flight of a million years, how weak would its telemetry signal be when received on earth? Use the PFD formula in the preceding problem.
13. SOLAR MAGNITUDE PROBLEM
Light, like the strength of radio signals, diminishes in intensity according to the inverse square law. Both light and radio signals are forms of electromagnetic radiation. They differ only in frequency and wavelength.
How much dimmer would be sun appear, relative to its appearance from earth, if you were on Mars (1.52 AU), Jupiter (5.2 AU), Saturn (9.58 AU), Uranus (19.17 AU), Neptune (30.03 AU), and Pluto (39.5 AU)?
14. SOLAR MASS PROBLEM
The sun blows off approximately a million tons of mass every second and has been doing so for its entire lifetime of about 4.5 billion years. Is this loss of mass significant or not?
Calculate the amount of mass which the sun has lost over the course of its lifetime and compare the result to the computed mass of the sun now. What percentage of the current mass of the sun does the loss represent? Does the loss seem significant?
15. INTERSTELLAR SPACE PROBLEM
Interstellar space is profoundly empty. One estimate is that it may have an average density of one hydrogen atom per cubic centimeter. That represents a vacuum infinitely better than anything achievable on earth. This problem helps to give you a sense of the scale involved.
Assume that the diameter of a hydrogen atom is 0.0000000001 meters and that the diameter of a tennis ball is 2.5 inches. If you scale up the hydrogen atom to the size of a tennis ball and imagine it suspended in the middle of a cube as large as a cubic centimenter scaled up to the same magnitude, how large would the cube be? (Give its dimensions in miles)
16.THE MARS PROBLEM
Celestial objects are quite small, explaining why telescopes are required for most of them.
Suppose you laid out a circle with a radius of one mile, with you at the center. How many quarters would you need to lay along the circumference of the circle for their combined diameters to be visually equivalent to the diameter of the planet Mars at its closest and farthest approaches to earth? Assume that Mars is approximately 3.5 arcseconds wide at its farthest approach and 25 arcseconds wide at its nearest approach to earth.