Fun With The HP 27S

Fun With The HP 27S

Notes:

*  The HP 27S is set to FIX 4

*  Since none of the formulas have trigonometric functions, they can also be programmed on the HP 17B and HP 17BII+

Partial Fraction Decomposition

(A ∙ x + B)÷((x + C) ∙ (x + D)) = R÷(x + C) + S÷(x + D)

Inputs:  A, B, C, D

Outputs:  R, S

Formula:

PARTFRAC2:(A+B+C+D)×0=IF(S(R):(B-C×A)÷(D-C)-R:(B-D×A)÷(C-D)-S)

Example:  (4 ∙ x + 3) ÷ ((x - 5) ∙ (x + 1))

A = 4, B = 3, C = -5, D = 1

Results:  R = 3.8333, S = 0.1667

Use that the 0×(var1+var2+...) to set an order of variables in the solver.  

2 x 2 Simultaneous Equations

A ∙ x + B ∙ y = C

D ∙ x + E ∙ y = F

The solutions are:

x = (C ∙ E - B ∙ F) ÷ (A ∙ E - B ∙ D)

y = (A ∙ F - C ∙ D) ÷ (A ∙ E - B ∙ D)

Formula:

SIM2X2:0×(A+B+C+D+E+F+L(M:A×E-B×D))=IF(S(X):(C×E-B×F)÷G(M)-X:(A×F-C×D)÷G(M)Y)

Example:  2x + 3y =5, -3x + 8y = -8

A = 2, B = 3, C = 5, D = -3, E = 5, F = -8

Results:  X = 2.5789, Y = -0.0526

Floor Function

floor(x):  the greatest integer less than or equal to x

floor(x):

If frac(x) = 0 Then return x

 Else If x≥0, then return intg(x) else return intg(x)-1

Formula:

FLOOR=IF(FP(X)=0:X:IF(X>=0:IP(X):IP(X)-1))

Examples:

X = 2.38, FLOOR = 2.0000

X = -9.21, FLOOR = -10.0000

Ceiling Function

ceil(x):  the least integer greater than or equal to x

ceil(x):

If frac(x) = 0 Then return x

  Else If x≥0, then intg(x)+1 else return intg(x)

Formula:

CEIL=IF(FP(X)=0:X:IF(X>=0:IP(X)+1:IP(X)))

Examples:

X = 2.38, CEIL = 3.0000

X = -9.21, CEIL = -9.0000

Rydberg Formula

The Rydberg formula measures the light's wavelength when an electron moves between energy quantum numbers (from higher to lower levels).  The Rydberg formula is to be used for simple atoms only, and is most used for hydrogen atoms.

1/λ = R ∙ Z^2 ∙ (1/n1^2 - 1/n2^2)

R = Rydberg's Constant ≈ 1.097373157 × 10^7 m^(-1)

Z = atomic number, 1 for hydrogen

n1, n2:  energy levels

Formula:

RYDBERG:INV(L)=1.09737316E7×SQ(Z)×(INV(SQ(N1))-INV(SQ(N2)))

N2 > N1

Example:

The energy of an hydrogen election from level 4 to level 2.  Z = 1

Z =1,  N1 = 2, N2 = 4

Result:  L = 4.8601E-7

Source:

Helmenstine, Todd. "What Is the Rydberg Formula and How Does It Work?" ThoughtCo, Aug. 28, 2020, https://www.thoughtco.com/what-is-the-rydberg-formula-604285   Retrieved November 4, 2021

Moment of Inertia - Circular Ring

D$OUT:  outside diameter

D$IN:  insider diameter.  For a circle, set D$IN = 0

I: moment of inertia

Formula:

I=PI×(D$OUT^4-D$IN^4)÷64

Examples:

Circular Ring:

D$OUT = 6.2, D$IN = 1.9;  I = 71.8935

Circle:

D$OUT = 6.2, D$IN = 0; i = 72.5332

Source:

"Properties Of Annual Sections" HP-19C/HP-29C Solutions: Civil Engineering  Hewlett-Packard.  1977

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Darts: The Probability of Getting the Best Score

Darts: The Probability of Getting the Best Score

In Darts, players throw darts at circular board.  The closer you are to the center of the circle, the more points you score.  Hitting the center of the circle (or close to it) scores you the bulls-eye.    

Today we will develop formulas that determine the chance of landing a dart in a specific scoring zone.  For this, we will assume that all the darts thrown will land in a specific area.  The dart that hits the cat who are unfortunate to be in its path don’t count.

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Area of a Circular Ring

We will be using the area of a circular ring heavily in this blog.  The general formula for it is:

 A =  π * ((outer radius)^2 – (inner radius)^2)

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Simplified Darts Board




 





This board has a circular area on a square board.  The square board has side length x.  There are three circular scoring areas:

(I)                 Bulls-eye circle with radius R1.  Obviously, this area has the most points.

(II)               Green scoring area, a circular ring with outer radius R2 and inner radius R1. This area is second best in points to the bulls-eye.

(III)             Yellow scoring area, a circular ring with outer radius R3 and inner radius R2. 

A dart that lands outside any of the scoring areas scores no points. 

Assuming all but a negligible of parts hits the square board, the chance that a dart hits the bulls-eye is:

P1 = (area of the bulls eye)/(area of the board) = π*R1^2/x^2

The probability that a dart hits the green scoring area:

P2 = (area of the green scoring area)/(area of the board) = π*(R2^2 – R1^2)/x^2

The probability that a dart hits the yellow scoring area:

P3 = (area of the yellow scoring area)/(area of the board) = π*(R3^2 – R2^2)/x^2

Finally, the probability of scoring anything at all on this board is:

P_any = P1 + P2 + P3 = π*P3^2/x^2

This assumes that the dart board is flat and each point of the dart board has an equal random chance.  Note that these probability formulas do not take into account the skill of the dart thrower. 

Standard Darts Board

 

 

The standard darts board is a circular board with radius R. Going from the inside out, we have:

 

(I)                 Inner Bulls-eye.  In the standard game, the inner bulls-eye is 50 points.  It is a circular area, with radius R1.  

(II)               Outer Bulls-eye – worth 25 points.  It is a circular ring, with outer radius R2 and inner radius R1. 

(III)             Inner Single Ring.  From here on out, the board is divided into 20 zones, with each zone assigned a different amount of points from 1 to 20 (not shown on the diagram).  Here I am only concerned with the circular ring in general.  This ring has the outer radius of R3 and inner radius of R2.

(IV)             Triple Ring.  Each of the zones in the triple ring are worth triple the points of designated zone.   Let’s assign the outer radius of the triple ring R4 and inner radius, R3.

(V)               Outer Single Ring.  With outer radius R5, and inner radius R6.

(VI)             Double Ring.  Each of the zones in the double ring are worth double the points.   This ring has the outer radius R6 and inner radius R5.

 

Here, I am concerned about hitting the general scoring areas. 

Probability of hitting a single scoring area:

P1 = (area of inner single ring + area of outer single ring)/(area of the board)

= (π * (R3^2 – R2^2) + π * (R5^2 – R6^2))/(π *R^2)
= (π * (R3^2 – R2^2 + R5^2 – R6^2)/(π * R^2)
= (R3^2 – R2^2 + R5^2 – R6^2)/R^2

Probability of hitting a double ring:

P2 = (area of the double ring)/(area of the board)

= (π * (R6^2 – R5^2))/(π * R^2)
= (R6^2 – R5^2)/R^2

Probability of hitting a triple ring:

P3 = (area of the triple ring)/(area of the board)

= (π * (R4^2 – R3^2))/(π * R^2)
= (R4^2 – R3^2)/R^2

Probability of hitting the bulls-eye:

PB = (area of the inner bulls-eye + outer bulls-eye)/(area of the board)

= (π * R1^2 + π * (R2^2 – R1^2))/(π * R^2)
= (π * R2^2)/(π * R^2)
= R2^2/R^2

Scoring 60

The highest a single dart can score is 60, hitting the 20 zone in the triple ring.  Note the triple ring is divided into 20 zones of equal area.  The required probability of scoring 60 points is:

P60 = (R4^2 – R3^2)/20 * 1/R^2 = (R4^2 – R3^2)/(20*R^2)

Knowing the area of the circular ring is key to calculating probabilities of hitting areas of the dart board.

Source:  http://en.wikipedia.org/wiki/Darts

Eddie

This entry is property of Edward Shore. 2014.