## Saturday, January 7, 2017

### HP 12C: Rule of 78, Slicing a Right Triangle, Sums, Projectile Motion

Announcement: Calculator Reviews

In the coming weeks, I have acquired a lot of calculators and plan to give a short review of each. They include, the original Hewlett Packard HP 10B, Casio EL-5500 III, and the Calculated Industries Construction Pro/Trig App.

Fun with the HP 12C (I lost count on how segments I done so far)

More fun with the HP 12C!  (The HP 12C is on my list of top ten calculators of all time - the other nine I have to think about... subject of a future post?).  I like using the HP 12C for a variety of applications and not just strictly finance.  Here are four more programs, enjoy!

HP 12C:  Rule of 78

When a mortgage, a loan, a lease, or other annuity is paid early, we can determine how much interest rebate is due by the Rule of 78:

Rebate = ( (n – k + 1) * FC ) / ( (n + 1) * n)

Where:
n = the length of the annuity (number of periods)
k = the period where the loan is paid off
FC = total interest, finance charge = PMT * n – PV

The program will require the user to input and compute the annuity variables [ n ], [ i ], [ PV ], and [PMT] ([FV] if a balloon payment is required).  Then enter the period # where the loan is paid off (k), and press [R/S].

Program:

Keep in mind: this is done on the HP 12C (regular).  For the HP 12C Platinum, the code for Last X is 43, 40 ([ g ] [ + ])

 STEP KEY CODE NUMBER 01 STO 1 44, 1 02 RCL PMT 45, 14 03 RCL n 45, 11 04 * 20 05 RCL PV 45, 13 06 + 40 07 CHS 16 08 STO 0 44, 0 09 RCL n 45, 11 10 RCL 1 45, 1 11 - 30 12 1 1 13 + 40 14 * 20 15 RCL 1 45, 1 16 ÷ 10 17 LST x 43, 36 18 1 1 19 + 40 20 ÷ 10 21 GTO 00 43, 33, 00

Example:  On a 48 month purchase of a \$20,000 car, financed at 5%, the purchaser pays the car off early after 24 payments (k = 24).  What is the rebate?

Output:
Find the payment:
[ f ] [X<>Y] (CLEAR FIN)  (if necessary)
48 [ n ]
5 [ g ] [ i ] (12÷)
20000 [ PV ]
[ PMT ]  (payment = -460.59)

24 [R/S]
Rebate:  \$87.84

Source:  Rosenstein, Morton.  Computing With the Scientific Calculator Casio: Tokyo, Japan.  1986.  ISBN-10: 1124161430

HP 12C: Slicing a Right Triangle

The program finds slices a right triangle into equal parts.  Using similar triangles, the bases and heights of similar triangles are found.

 STEP KEY CODE NUMBER 01 RCL 0 45, 0 02 RCL 2 45, 2 03 ÷ 10 04 INTG 43, 25 05 STO 3 44, 3 06 1 1 07 STO 4 44, 4 08 RCL 0 45, 0 09 RCL 3 45, 3 10 RCL 4 45, 4 11 * 20 12 - 30 13 R/S 31 14 RCL 1 45, 1 15 * 20 16 RCL 0 45, 0 17 ÷ 10 18 R/S 31 19 1 1 20 STO+ 4 44, 40, 4 21 RCL 2 45, 2 22 RCL 4 45, 4 23 X≤Y 43, 34 24 GTO 08 43, 33, 08 25 GTO 00 43, 33, 00

Input:  Pre-store the following values:
Run in Register 0 (R0)
Rise in Register 1 (R1)
Number of partitions in Register (R2)

Output:  Loop:
Base of the smaller triangle (x), press [ R/S ]
Height of the smaller triangle (y), press [ R/S ]
Loop ends after n pairs

Example:  Run = 5  (R0), Height = 3 (R1), Number of Partitions = 5 (R2)

Output:
 X 4 3 2 1 0 Y 2.4 1.8 1.2 0.6 0

HP 12C:  Sums of Î£x, Î£x^2, Î£x^3

This program takes two arguments:

Y: x
X: n  (where n=1, n=2, n=3)

If n = 1, the sum Î£ x from 1 to n is calculated
If n = 2, the sum Î£ x^2 from 1 to n is calculated
If n = 3, the sum Î£ x^3 from 1 to n is calculated

If n is not 1, 2, or 3, an error occurs.

Program:

Keep in mind: this is done on the HP 12C (regular).  For the HP 12C Platinum, the code for Last X is 43, 40 ([ g ] [ + ]) and the step numbers are three digits (000 instead of 00).

 STEP KEY CODE NUMBER 01 X<>Y 34 02 STO 1 44, 1 03 X<>Y 34 04 STO 0 44, 0 05 1 1 06 - 30 07 X=0 43,35 08 GTO 21 44, 33, 21 09 RCL 0 45, 0 10 2 2 11 - 30 12 X=0 43, 35 13 GTO 29 44, 33, 29 14 RCL 0 45, 0 15 3 3 16 - 30 17 X=0 43, 35 18 GTO 46 43, 33, 46 19 0 0 20 ÷ 10 21 RCL 1 45, 1 22 ENTER 36 23 * 20 24 LST X 43, 36 25 + 40 26 2 2 27 ÷ 10 28 GTO 00 43, 33, 00 29 RCL 1 45, 1 30 ENTER 36 31 * 20 32 LST X 43, 36 33 X<>Y 34 34 3 3 35 * 20 36 + 40 37 RCL 1 45, 1 38 3 3 39 Y^X 21 40 2 2 41 * 20 42 + 40 43 6 6 44 ÷ 10 45 GTO 00 43, 33, 00 46 RCL 1 45, 1 47 ENTER 36 48 ENTER 36 49 1 1 50 + 40 51 * 20 52 ENTER 36 53 * 20 54 4 4 55 ÷ 10 56 GTO 00 43, 33, 00

Example:  n = 5
Y: 5, X: 1. Result: 15
Y: 5, X: 2. Result: 55
Y: 5, X: 3. Result 225

HP 12C:  Projectile Motion, No Air Resistance:  Maximum Distance (U.S. Units)

For an object that travels in a projectile motion, we can track its range (distance traveled from the beginning) and height by:

R = v^2 * sin (2 * Î¸)/g
H = (v^2 * (sin Î¸)^2) / (2 * g)

Where:
v = initial velocity
Î¸ = initial angle
g = Earth’s gravity.  For in US units, g = 32.1740468 ft/s^2.
This program uses the approximation g ≈ 32.174 ft/s^2

The projectile will have maximum range (distance) if we aim the object at 45°.
--------------------------
Aside:  Why?

Let’s let range (R) be a function of angle (Î¸):

R = v^2/g * sin(2 * Î¸)

Find the critical points by finding the zero of the first derivative:

dR/dÎ¸ = 2 * v^2/g * cos (2 * Î¸)
0 = 2 * v^2/g * cos (2 * Î¸)
0 = cos (2 * Î¸)
arccos 0 = 2 * Î¸
Ï€/2 = 2 * Î¸
Î¸ = Ï€/4

Now we can use the second derivative to test whether the function is at a maximum (less than 0) and minimum (more than 0) at the crucial point.

d^2 R/dÎ¸^2 = -4 * v^2/g * sin(2 * Î¸)
Let Î¸ = Ï€/4
-4 * v^2/g * sin(2 * Ï€/4) = -4 * v^2/g * sin(Ï€/2) = -4 * v^2/g < 0
(We are assuming the initial velocity is positive, and g ≈ 32.174 >0)

Since the second derivative at Î¸ = Ï€/4 is negative, the range is at its maximum.

Note that in calculus, angles are measured in radians.  Ï€/2 radians in degrees is 90° and Ï€/4 radians in degrees is 45°.  (We are only concentrating on angles between 0° and 90°)
--------------------------

To find the maximum range and height, substitute at Î¸ = 45° and range and height are:

R = v^2 /g
H = v^2 / (4 * g)
The time this certain projectile lasts is:

T = (v * √2) / (2 * g)

Program:

Keep in mind: this is done on the HP 12C (regular).  For the HP 12C Platinum, the code for Last X is 43, 40 ([ g ] [ + ])

 STEP KEY CODE NUMBER 01 STO 1 44, 1 02 2 2 03 ÷ 10 04 LST x 43, 36 05 √x 43, 21 06 * 20 07 3 3 08 2 2 09 . 48 10 1 1 11 7 7 12 4 4 13 STO 0 44, 0 14 ÷ 10 15 R/S 31 16 RCL 1 45, 1 17 ENTER 36 18 * 20 19 RCL 0 45, 0 20 ÷ 10 21 R/S 31 22 4 4 23 ÷ 10 24 GTO 00 43, 33, 00

Input:  velocity in ft/s   (convert from mph to ft/s by multiplying it by 22/15)

Output:
time of projectile in seconds, [R/S]
range of projectile in feet, [R/S]
height of projectile in feet

Example:
V = 25 mph = 36.6666667 ft/s  (110/3)

Output:
Time:  0.81 sec,  Range: 41.79 ft,  Height: 10.45 ft

Source:  Rosenstein, Morton.  Computing With the Scientific Calculator Casio: Tokyo, Japan.  1986.  ISBN-10: 1124161430

Eddie

This blog is property of Edward Shore, 2017.  (2017, wow!  7 days already have passed.)

#### 4 comments:

1. Very useful information and program instructions with HP 12C calculator.
Keep the good work !!

1. Thank you! I love working with the 12C. I wasn't a fan at first (we're talking 20 years ago), but it grew on me. It's one the of the best calculators.

Eddie

2. Many thanks Eddie once again for a superb article on programming a calculator. Would love to read about your top ten all-time-favorite calculators! I think it would be interesting to see which company's calculators tug at your heart (and head) the most.
TI
HP
Casio

Thanks again for all your posts and your incisive videos on YouTube also.

Best wishes.
Joseph

1. Thank you, Joseph. I have made not made the list yet. I have four definite calculators so far.

Eddie

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