University of Houston math contests
07-11-2019, 01:12 AM (This post was last modified: 07-11-2019 01:50 AM by telemachos.)
Post: #1
 telemachos Junior Member Posts: 17 Joined: Apr 2017
University of Houston math contests
Held each year, intended for high school students, they include calculator exams.

These HP calculators are now permitted (early on, only TI calculators were permitted):

HP-9G
HP-28 series
HP-38G
HP-39 series
HP-40 series
HP-48 series
HP-49 series
HP-50 series

<http://mathcontest.uh.edu/>

Forum members may find this trove of problems amusing, trivial, or (in some cases) worthwhile education in the use of their HP calculator. (Not surprisingly, the HP-50 has some distinct advantages over the other permitted models.)
07-11-2019, 03:30 PM
Post: #2
 telemachos Junior Member Posts: 17 Joined: Apr 2017
RE: University of Houston math contests
For example, I found this problem, from the 2019 exam, worthwhile:

A quadrilateral has its vertices (listed in clockwise order) as

(–6.94, –1.2), (–5.16, 3.84), (3.4, 2.18), and (2.46, –6.72).

Give the area of this quadrilateral.

Surveyors among you may recognize this one. A short RPL program on the 50g solves it quickly.
07-11-2019, 08:26 PM
Post: #3
 John Keith Senior Member Posts: 1,032 Joined: Dec 2013
RE: University of Houston math contests

07-11-2019, 10:36 PM
Post: #4
 Albert Chan Senior Member Posts: 2,682 Joined: Jul 2018
RE: University of Houston math contests
(07-11-2019 03:30 PM)telemachos Wrote:  (–6.94, –1.2), (–5.16, 3.84), (3.4, 2.18), and (2.46, –6.72).

Give the area of this quadrilateral.

Another way is to calculate angles θ between 2 diagonals:

Slope m1 = (2.18 + 1.2) / (3.4 + 6.94) = 3.38 / 10.34
Slope m2 = (-6.72 - 3.84) / (2.46 + 5.16) = -10.56 / 7.62

tan(θ) = (m1-m2) / (1 + m1 m2) ≈ 3.13114
sin(θ) = tan(θ) / √(1 + tan(θ)²) ≈ 0.952598

area = ½ product of diagonals * sin(θ) ≈ ½ √(3.38² + 10.34²) * √(10.56² + 7.62²) * 0.952598 = 67.473
07-12-2019, 10:34 AM (This post was last modified: 07-12-2019 12:09 PM by Albert Chan.)
Post: #5
 Albert Chan Senior Member Posts: 2,682 Joined: Jul 2018
RE: University of Houston math contests
(07-11-2019 08:26 PM)John Keith Wrote:  It's already done for you.

For quadrilateral, we could reduce ops with a more compact formula.

Area = ½|(x3-x1)(y4-y2) - (x4-x2)(y3-y1)|

This reduce Shoelace 9 multiply and 8 add/sub to just 3 multiply and 5 subtract.

We could also use this formula for triangle area, by letting {x4,y4} = {x3,y3}

Example, with XCas:

area4(a,b,c,d) := 0.5*abs(det([c-a,d-b]))
area3(a,b,c) := area4(a,b,c,c)

a,b,c,d := [-6.94,-1.2], [-5.16, 3.84], [3.4,2.18], [2.46,-6.72]

area4(a,b,c,d) → 67.473
area3(a,b,c) + area3(a,c,d) → 23.0486 + 44.4244 = 67.473
07-12-2019, 04:41 PM
Post: #6
 telemachos Junior Member Posts: 17 Joined: Apr 2017
RE: University of Houston math contests
I think this makes the best use of the HP-50g's capabilities:

Input:

{ [–6.94 –1.2] [–5.16 3.84] [3.4 2.18] [2.46 –6.72] }

Program (56.5 bytes):

<< DUP HEAD + << CROSS >> DOSUBS ΣLIST ABS 2 / >>

Output: 67.473
07-12-2019, 11:33 PM (This post was last modified: 07-12-2019 11:35 PM by RMollov.)
Post: #7
 RMollov Member Posts: 264 Joined: Dec 2013
RE: University of Houston math contests
(07-12-2019 04:41 PM)telemachos Wrote:  I think this makes the best use of the HP-50g's capabilities:

Input:

{ [–6.94 –1.2] [–5.16 3.84] [3.4 2.18] [2.46 –6.72] }

Program (56.5 bytes):

<< DUP HEAD + << CROSS >> DOSUBS ΣLIST ABS 2 / >>

Output: 67.473

... and the one I've been using in my survey package for 20+ years on HP48G
07-13-2019, 01:15 PM
Post: #8
 John Keith Senior Member Posts: 1,032 Joined: Dec 2013
RE: University of Houston math contests
(07-12-2019 04:41 PM)telemachos Wrote:  I think this makes the best use of the HP-50g's capabilities:

Input:

{ [–6.94 –1.2] [–5.16 3.84] [3.4 2.18] [2.46 –6.72] }

Program (56.5 bytes):

<< DUP HEAD + << CROSS >> DOSUBS ΣLIST ABS 2 / >>

Output: 67.473

Very nice program! I think it is the smallest HP-48 compatible shoelace program yet. It can be made even shorter using the null-tag method from this thread, which shaves 9 bytes off the size, like so:

Code:
 \<< DUP HEAD + :: CROSS DOSUBS \GSLIST ABS 2 / \>>

07-13-2019, 03:57 PM
Post: #9
 DavidM Senior Member Posts: 989 Joined: Dec 2013
RE: University of Houston math contests
(07-12-2019 04:41 PM)telemachos Wrote:  ...
I think this makes the best use of the HP-50g's capabilities:

Input:

{ [–6.94 –1.2] [–5.16 3.84] [3.4 2.18] [2.46 –6.72] }
...

Getting the data into that format could be an awkward exercise if attempting to enter it directly from the calculator's keyboard. It's very close, though, to the format that would exist if you used the built-in MatrixWriter to create it:
Code:
[[–6.94 –1.2] [–5.16 3.84] [3.4 2.18] [2.46 –6.72]]

On both the 48GX and the 50g, the following two commands will convert a matrix created by the MatrixWriter to a "list of vectors" format compatible with the given shoelace programs in this thread:
Code:
→ROW →LIST

I find the MatrixWriter to be convenient for this type of data entry (as well as maintenance). Judicious use of the various array commands (and AXL on the 50g) can often help to transform an array to a wide variety of useful formats.
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