Problem-solving Method

The following is a method of solving problems that applies to the problem corner in this magazine as well as problems surveyors face each day. This method involves six specific steps followed by a description of each step in the process and an example.

Professional surveyors encounter the same kinds of problems with a change in numbers; it is, therefore, efficient and time saving to be able to derive an analytic or general solution to the problem first and then enter new sets of data as needed. Many of my problems in the Problem Corner will test your ability to do just that, to get an analytic or general solution.

The steps are as follows:
  1. Given
  2. To Find
  3. Basic Principles
  4. Pertinent Equations
  5. Analytic Solution
  6. Numerical Solution
Not all problems will require every step; however, these six steps should handle most problems.

1. Given: Answer the following: Does the problem contain givens in the form of whole numbers, numbers with significant figures, angles, letters, shapes, etc.? If so, the procedure is to assign a letter or symbol to each given, because the Analytic Solution must contain only these.

2. To Find: This is what the problem seeks to determine. Is it about solving for a distance, an angle, a direction, a shape, etc.? Again, a letter or symbol is needed. The Analytic Solution must be that letter or symbol in terms of the letters or symbols in the Given. The solution must be expressed in simplest terms.

3. Basic Principles: List the principles that relate to all the Given and the To Find letters or symbols.

4. Pertinent Equations:
These are the equations that pertain to the Basic Principles that are used to find the relation between the quantities.

5. Analytic Solution: The Pertinent Equations are used to solve for the To Find quantity. The solution of the To Find must be reduced to first degree.

This step provides the general relationships between all of the quantities and allows us to determine how a change in one variable will affect the others. It enables us to scale and/or vary the problem based on the Givens. This step completes the difficult portion of the analysis.

6. Numerical Solution:
Lastly, substitute the numbers for the Givens to yield the Numerical Solution. Pay careful attention to the consistency required in terms of significant figures.

Let’s look at an example.

A surveyor is called in to measure a farmer’s well. A measurement of its depth is required. (Let’s assume that the latest lidar equipment is not working.) You could lower the farmer in a bucket attached to a knotted rope and measure that way. Or you could use a little basic physics.

Any object dropped near the surface of the Earth is acted upon by gravity and will fall under that acceleration. Another important aspect of real problem solving is to recognize that certain assumptions have to be made. No one can eliminate all error, but it is good to be able to assign an acceptable degree of precision. We assume here that there is no air resistance or that the falling object is so heavy that air viscosity effects are negligible.

Here goes. This is a Uniform Accelerated Motion problem and involves five quantities: initial velocity, final velocity, acceleration, distance, and time. The trained professional knows that you need to be given any three of these quantities to solve for the remaining two.

To start the method, assign a letter to each of the five quantities: v, vi, a, s, and t:
v = the final velocity, vi = the initial velocity, a = the acceleration, s = distance, and t = time.

Drop a ball or solid object from the mouth of the well and measure the time interval to the splash. (You’ll need to conduct several trials to yield the average).

So, you know “a,” you will measure “t,” and unless you can find another quantity (either vi or v), you cannot solve this problem. You seem to be missing a vital piece of information.

This is good: There are problems that cannot be solved because there is just not enough pertinent information. A professional needs to know this.

But, wait a minute. If we read and analyze the problem very carefully, we notice that the ball is dropped, which means that its initial velocity (vi) = 0. This is the vital third piece of information. We can now proceed to apply this method.

1. Given: vi = 0, a = g = acceleration due to gravity, and t will be measured. (The measurement gives a value for t = 8.1 seconds.)

2. To Find: s

3. Basic Principle:
Uniform Accelerated Motion

4. Pertinent Equations: There are several uniform accelerated motion equations, but we want to select the one that contains only the Given and the To Find. (This is how we select from the many equations available under the Basic Principle).

The equation is: s = vit + ½ a t2

5. Analytic Solution:

vi = 0, and thus, without any math manipulation

s = ½ g t2

This is our analytic, or general, solution. It applies to the surface of the Earth, on top of a mountain where g is different (or on another planet, so long as its g is known). This analytic relationship also allows us to solve for s, g, or t, given the other two quantities.

6. Numerical Solution:

s = ½ 9.8(m/s2)(8.12 sec2) = 321.5 m (a deep well)

Please note that if we knew the depth and needed to solve for t because vi = 0 (we don’t have to solve a quadratic equation), we get that t2 = 2s/g and therefore t = (2s/g)½

Here are some things to look for in my problems.

  1. If a geometric problem has only whole number angles, that is a signal that any unknown angle follows suit and must also be a whole number.
  2. When no lengths are given, then the solution must use letters for the lengths to yield an analytic or general solution.
  3. There can be more than one method of solution if conditions 1 and 2 are obeyed.
  4. Significant figures are important but must remain within the spirit of the problem. 
  5. Some of the forthcoming problems will require a little calculus so that only analytic relationships will do.
Specifically relating to Problem #231: All of the angles shown are in whole numbers, so a whole number angle is implied for the solution. No lengths are shown. The triangle is therefore of general size, indicating that an analytic solution is implied.

Thanks for the years of support; these problems are fun.

About the Author

  • Benjamin Bloch, PhD
    Benjamin Bloch, PhD
    Benjamin Bloch's many accomplishments include having been a violinist with the New York Jr. Symphony Orchestra, traveling to the North Polar Region for the Office of Naval Research, working as Staff Physicist at the Grumman Aerospace Corporation, earning a Ph.D. in theoretical physics Polytechnic Institute of Brooklyn, and teaching every level of college and high school math and physics. He is also an author and contributes to the Problem Corner of this magazine.

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