Math, engineering, trades: steepness measurement: definition of grade

Tutoring math, you encounter grade. The tutor discusses its definition and why it might be surprising.

Grade is defined as 100%*(vertical/horizontal). In the above diagram, it would be as follows:


By itself, rise/run is called slope.



What follows is a distinction that, to me, is important and interesting:

grade is not

rise/distance traveled

since, of course, you can’t drive along the horizontal course of a hill; rather, you can only drive on its surface.

At level, grade and (rise/distance traveled) are both zero. They remain virtually the same even at 20% grade, when (rise/distance traveled) is 19.6%. As the grade increases, however, they differ dramatically: at 100% grade, (rise/distance traveled) is 70.7%.

My interest in the difference between grade and (rise/distance traveled) is philosophical: why base a value on an indirect measurement (horizontal distance), when a direct measurement (distance traveled) is available?

In math, we use slope, of course; however, it’s usually in a context where actual measurements aren’t used. Rather, it’s just on paper.


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculator usage: the ENG feature on the Casio fx-991ES PLUS C

Tutoring math, calculator features are always topical. The tutor mentions the ENG function on the Casio fx-991ES PLUS C.

Back in my Halloween, 2015 post, I explain the general idea of the ENG function – that engineers and technicians often like to refer to measurements in “thousand” groupings. For instance, 1249W is 1.249kW; 0.067A is 67mA.

Example: convert 0.78A to mA

  1. Key 0.78 = (you must press = for this to work)
  2. Press ENG: hopefully you see 780×10-3

Example: convert 1249W to kW.

Solution: to go up to the bigger unit, use SHIFT ENG, like so:

  1. Key 1249 = (once again, you must press =)
  2. Press SHIFT then ENG. Hopefully you see 1.249×103


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

AC Electricity, electronics: series high-pass filter

The tutor gives background along with a basic explanation of how a high-pass filter works.

A high-pass filter will send along high frequency signals but block low frequencies. It can do so because the impedance, Xc, of a capacitor of capacitance C, at frequency f, is


At very high frequency, therefore, Xc≈0

The total impedance, Z, of a resistor R and capacitor C in series is given by

Z=(R2 + Xc2)0.5

Therefore, with input voltage V, and instantaneous current i, Ohm’s Law gives

V=i(R2 + Xc2)0.5

The voltage across the resitor is


Therefore, the ratio of the voltage across the resistor to the source is

VR/V = iR/i(R2 + Xc2)0.5 = R/(R2 + Xc2)0.5

As mentioned earlier, for high frequency, Xc≈0, giving

VR/V ≈ R/(R2)0.5 = 1

With a high-pass filter, the maximum output voltage for a high frequency signal equals the input voltage.

Convention says that the critical frequency, fc, is that at which VR/V = 1/(2)0.5 = 0.707, which occurs when Xc = R:

VR/V = R/(R2 + Xc2)0.5 = R/(R2 + R2)0.5 = R/(2R2)0.5

which leads to

VR/V = 1/20.5 = 0.707

Therefore, a high-pass filter will pass any frequency higher than the critical frequency fc, where fc is calculated from

Xc = R

1/(2πfcC) = R

1 = 2πfcCR

1/(2πCR) = fc

By that reasoning, a 10kΩ resistor, in series with a 12pF (picoFarad) capacitor, placed in series, should produce a high-pass filter with critical frequency fc = 1.3MHz. The output would be read across the resistor R.


Serway, Raymond A. Physics for Scientists and Engineers with modern physics. Toronto: Saunders College Publishing, 1986.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculator usage, physics: εο: permittivity of free space constant on the Sharp el-520w

The tutor tells how to access the built-in εο constant on the Sharp el-520w.

The Sharp el-520w has 52 built-in constants relating to physics, chemistry, etc.

Here’s how to call up εο, which has value 8.85×10-12:

  1. Press the CNST key, and you’ll be asked which of constants (01-52) you require.
  2. Key in 13. You’ll see 8.854187817×10-12 appear.


Sharp Scientific Calculator Model EL-520W Operation Manual.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Physical chemistry: efficiency of internal combustion engine in hot vs cold weather

The tutor examines the idea that internal combustion engines are more efficient in cold weather.

An upper limit for efficiency of an internal combustion engine is

eff = (Tcombust – Tsurrounding)/Tcombust


Tcombust is the temp of the combustion cylinder

Tcombust, Tsurrounding both in degrees Kelvin (Celsius + 273).

Let’s imagine a diesel engine, whose average internal cylinder temperature might be around 1600°C. Then at outdoor temp of 25°C (293K) we have

eff = (1873 – 298)/1873 = 84.1%

Likewise, at outdoor temp -25°C we have

eff = (1873 – 248)/1873 = 86.8%

The 2.7% increase in efficiency at -25°C vs 25°C may be noticeable to an operator.


White, J. Edmund. Physical Chemistry: College Outline Series. New York: Harcourt Brace Jovanovich, 1987.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Lifestyle: toy repair with J-B Weld, part II

The tutor continues about a toy repair, with some reinforcement ideas.

While we were repairing the toy a few nights ago (see my previous post), my father-in-law suggested that, after the first repair cured, a second application should be considered around the outside. Such reinforcement, he commented, would give the repair its best chance of holding.

I considered his counsel from an engineering point of view: how much extra strength could we anticipate from application of J-B Weld around the outside of the repair site?

Let’s imagine the shear force to be straight forward. The strength of a reinforcement can be, generally, proportional to its left-right length multiplied by its height, then by the square of its forward length. Assuming the J-B Weld works as an integral piece after drying, I imagined an outside application along each side plane. The application would be about 20 times the height of the original shear, then its same forward length, but only about 1/30 of its left-right length. Compared with the first repair reuniting the two sheared surfaces, the reinforcement strength per side might be 20(1/30) or 2/3. Both sides together could offer reinforcement strength of 2(2/3)=4/3 or 1.33 times the strength of the original repair, more than doubling its shear resistance.

With these numbers in mind, I took my father-in-law’s advice and made the reinforcement application about 24 hours ago. The repair should be ready right now.

I’ll be sharing more about this fascinating toy repair:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculator usage: The ENG function on the Casio fx-260solar

The tutor explains his recent understanding of a function he’s wondered about.

I’ve noticed the ENG function on more than one calculator, but have never used it. I’ve always assumed it means “engineering”; since I’m not one, it makes sense that I’m unfamiliar with it.

Yesterday my curiosity finally focused on this mysterious ENG function. You access it by SHIFT ÷ on the Casio fx-260solar. If you’re in COMP mode (I haven’t tried it with other modes), it seems to change the entered number to the highest power of 103 for which the number will be > 1. Examples:

0.056 SHIFT ÷ gives 56×10-3

0.000362 SHIFT ÷ gives 362×10-6

12037059.1 SHIFT ÷ gives 12.0370591×106

While I’m not an engineer, this notation is familiar to me. I know that in electronics, it’s common to refer to 0.056A as 56mA. Similarly, 0.000178A will likely by referred to as 178µA, also known as 178×10-6A. 12400000Ω would likely be referred to as 12.4MΩ (M=Mega=106).

I have more to say about the ENG function:)


Casio fx-260solar operation manual. London: Casio Electronics Co., Ltd.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.