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	4.00	3.00	2.00	1.00	0.00
Y	2.40	1.80	1.20	0.60	0.00

 

HP 12C:  Sums of Σx, Σx^2, Σx^3

This program takes two arguments:

Y: x

X: n  (where n=1, n=2, n=3)

If n = 1, the sum Σ x from 1 to n is calculated

If n = 2, the sum Σ x^2 from 1 to n is calculated

If n = 3, the sum Σ 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 * θ)/g

H = (v^2 * (sin θ)^2) / (2 * g)

Where:

v = initial velocity

θ = 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 (θ):

R = v^2/g * sin(2 * θ)

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

dR/dθ = 2 * v^2/g * cos (2 * θ)

0 = 2 * v^2/g * cos (2 * θ)

0 = cos (2 * θ)

arccos 0 = 2 * θ

π/2 = 2 * θ

θ = π/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/dθ^2 = -4 * v^2/g * sin(2 * θ)

Let θ = π/4

-4 * v^2/g * sin(2 * π/4) = -4 * v^2/g * sin(π/2) = -4 * v^2/g < 0

(We are assuming the initial velocity is positive, and g ≈ 32.174 >0)

Since the second derivative at θ = π/4 is negative, the range is at its maximum.

Note that in calculus, angles are measured in radians.  π/2 radians in degrees is 90° and π/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 θ = 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.)