Math Formulas for the Scientific Calculator

I realize that probably 9 out of 10 people consider math and mathematical formulas to be one of the most boring subjects, but this is why we have pocket scientific calculators!

So with that in mind...

Frustrum of a Cone:
V = ((h*pi)/3)*(r2 + R2 + (r*R))
h = height
r = the radius of one end (diameter / 2)
R= the radius of the opposite end.
Pi = 3.141592654.
A frustrum of a cone is simply a cone with the top sliced off. Turn it over and it becomes the shape of a common drinking glass.
This is the formula that will tell you how much liquid the glass will hold.
Note: If measurements are taken in centimeters, the result will be in milliliters, which can be easily converted to fluid ounces by multiplying by .033814.
Cylinder:
V = pi*r2*h
Pi = 3.141592654
r = radius (diameter / 2)
h = height.
This is of course the volume of any cylindrical object such as a tall glass, kitchen canisters, trash cans, etc. I once calculated the volume of ink in a fountain pen ink cartridge, a surprisingly small amount.
I've also used this formula to get an approximate idea of the amount of water capacity in a water tank...if you know the circumference (which I measured by foot), than the diameter is the circumference divided by pi.
The capacity was around 1.5 million gallons = 5.7 million liters = 48,000 Barrels!
Distance of Fall (in feet):
D = 16 * T2
T = time (duration of fall) in seconds.
This formula will tell you how deep a well is or how high the cliff is that you are standing on by dropping a stone and timing the length of the fall in seconds.
Ignored here is the effect of air resistance, which is negligible for a small distance and a dense object such as a stone.
Also left out is the time it takes for the sound to reach your ears...which gives us a second formula,
Distance of fall (2):
D = 16 * (T - ((16 * T) / 1100))2
T = time (duration of fall) in seconds.
This formula automatically corrects for the speed of sound, estimated at approximately 1100 feet per second.
Acceleration from Gravity:
S(fps) = T * (32.173 - cos(L * 2))
T = time (duration of fall) in seconds
L = latitude
This very precise formula will give you the speed of the rock (in feet per second) upon impact, taking into account the latitude at the location of the fall.
Because the earth is not a perfect sphere, acceleration varies according to latitude by as much as one half of one percent.
To convert feet per second to miles per hour, multiply the result by .6818.
Speed of Sound in Water:
s = 47.55.2 + 15.067(C) - .1765(C2) + .00085(C3)
s = Feet per second
C = Water Temperature in Degrees Celsius
Unfortunately this formula only applies to fresh water.

For salt water...

Speed of Sound in Salt Water:
s = 47.55.2 + 15.067(C) - .1765(C2) + .00085(C3) + 3.9(p - 35) + (d / 61)
s = Feet per second
C = Water Temperature in Degrees Celsius
p = salinity in parts per thousand
d = water depth in feet
I unfortunately have no idea how to measure salinity!
Finding a Mach Number:
1. a = (cas / 661.5)2
2. b = ((a / 5) + 1)3.5 - 1
3. c = palt * 6.875 * .000001
4. d = (1 - c)-5.2656
5. e = ((((d * b) + 1).286) - 1) * 5
6. Mach Number = sqr(e)
cas = calibrated airspeed
palt = pressure altitude
sqr = square root
This is a rather complex computation because the speed of sound actually varies with altitude.
The scientific calculator's powers key comes in pretty handy here.
Example:
calibrated airspeed = 350 knots
pressure altitude = 25,500 feet
a = (350 / 661.5)2 = .2799
b = ((.2799 / 5) + 1)3.5 - 1 = .2101
c = 25500 * 6.875 * .000001 = .1753
d = (1 - .1753)-5.2656 = 2.7592
e = ((((2.7592 * .2101) + 1).286) - 1) * 5 = .6984
The Mach Number is the square root of e: .8357
In this example the plane is flying at a little more than eight-tenths of the speed of sound.
Note: The calculations in this example have been rounded off to four decimal places.