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
4.00
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.)  







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