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falzoon
Recreations: φ = (√5 + 1)/2 sin(9º) = cos(81º) = (1/2)√[2 - √(φ + 2)] sin(18º) = cos(72º) = (φ - 1)/2 sin(27º) = cos(63º) = (1/2)√[2 - √(3 - φ)] sin(36º) = cos(54º) = (1/2)√(3 - φ) sin(54º) = cos(36º) = φ/2 sin(63º) = cos(27º) = (1/2)√[2 + √(3 - φ)] sin(72º) = cos(18º) = (1/2)√(φ + 2) sin(81º) = cos(9º) = (1/2)√[2 + √(φ + 2)] A delightful corollary: sin(54º) = sin(18º) + 1/2 Quick solutions to: Asin(x) # Bcos(x) = C (where # = + or -) x = 2nπ + 2*arctan{ [A ± √(A^2 + B^2 - C^2)] / (C # B) } (for integral n) Yields two solutions for each n.
WolframAlpha Table question?
On entering Table[log(n), {n, 3, 5}], WolframAlpha gives
only the exact values of log(3), log(4), log(5).
How do I get it to give the approximate values instead?
4 AnswersMathematics5 years ago
How to cut the cake?
A rectangular slice of orange cake that has been lying around for too long, has
developed an elliptical area of mold that extends in similar manner, directly down
through the whole layer. Fortunately, the mold does not produce diffusible toxins,
so if all the mold is cut out, the rest of the cake will be edible.
Where would you make two intersecting, straight cuts, that each cut the X-axis,
so that all the mold is removed, but also removing as little cake as possible?
This creates a triangular area around the ellipse, with one triangle side along the
X-axis, such that the total of the orange areas inside the triangle is a minimum.
Of course, all three triangular sides must be tangent to the ellipse.
The equation to the ellipse is : [y + x - √(5/2)]^2 / 8 + [y - x - √(5/2)]^2 / 2 = 1.
(The diagram was produced from the 'familiar' horizontal major axis ellipse :
x^2/a^2 + y^2/b^2 = 1, with a = 2 and b = 1, but rotated 45º anticlockwise and
then shifted up by √(5/2) so that it is tangent to the X-axis.)
1 AnswerMathematics6 years ago
What is the answer to this least distance problem?
Sunny always reckoned he had been misnamed because he prefers the shade.
One particularly hot day he reluctantly went to a birthday party for Honey, his
bitter and twisted sister, where he had to cross a field dotted with trees.
The diagram shows their houses represented as red squares, sunlit areas as
yellow squares and shade under trees as numbered, green squares.
Imagining Sunny as being a point, that only moves in straight lines, he is
allowed to move in any direction and able to cross any of the squares.
If he reaches the edge or corner of any green square, then he is in the shade.
Likewise, on any side or corner of a red square, he is deemed to be in a house.
Each of the squares has a side of 1 unit.
Question: Walking at a constant speed, what distance does he cover over
sunlit areas, such that he spends the least time in the sun?
This is a fairly simple puzzle that manually, shouldn't take too long. I'd be
interested in knowing of an algorithm (for any number of green squares), that
solves the puzzle by either brute force or something more sophisticated.
1 AnswerMathematics6 years ago
Wondering what are the properties of this surface and the length of the diagonals?
In my doodlings I drew this object, but I'm lacking in knowledge of how to describe
the top surface (coloured). What type of distortion is this and does this particular
surface have a name? Where can I read more about this type of surface?
Further to this, how are the values of the diagonals of the surface calculated?
In trying to construct a 3D model from a cardboard tissue box, using cellophane or
paper for the surface, was difficult. I then found the best method was to wrap cotton
thread around the box, from side AE to side BF, at equal intervals (1 cm) along
side AE, always keeping the thread parallel to AB. Only then, looking down from
the top, was I sure that diagonal AF is concave down, while BE is concave up.
I think it's a hyperbolic surface, but I also think that the math may be too difficult,
so any more than simple algebra or simple calculus may be out of my depth.
If the ends (square and triangle) were changed to other shapes, I can see all sorts
of variations and complications, although between any two points that could be
labelled as a diagonal, I imagine the math would be much the same (?).
1 AnswerMathematics7 years agoHow can I prove that (3*10^n - 2) is divisible by 17 for some integer n ...?
... and find n by simpler means than trial and error on testing with n = 1, 2, 3, etc.?
2 AnswersMathematics7 years agoFind the square with internal triangle of minimum area?
Draw a triangle inside a square such that the vertices of the triangle touch three
different sides (not the corners) of the square. If all line segments are different
integers, then what is the value of the side of the square if the triangle is of
minimum area?
Is this related to my previous problem? -
3 AnswersMathematics7 years agoFind the square with internal quadrilateral of minimum area?
Draw a convex quadrilateral inside a square such that each vertex of the quadrilateral
touches a different side of the square. If all line segments are different integers, then
what is the value of the side of the square if the quadrilateral is of minimum area?
2 AnswersMathematics7 years agoLines and Angles Problem. Can a circuit be completed?
Please see diagram at: http://s205.photobucket.com/user/falzoon/media/Lin...
This problem that I haven't solved yet, and not likely to, has plagued me for decades.
Starting at the origin, straight lines, a_1, a_2, a_3, ..., a_n, of equal length (1 unit),
are placed end to end, so that a_1 makes a 1º angle with the X-axis, a_2 makes a
2º angle with a_1 produced, a_3 makes a 3º angle with a_2 produced, etc.
Angles are always made to the same side so that the lines turn in the same
'circular' direction.
Questions:
1) Is there a line, a_n, that eventually meets a_1 at any point on it? If so, what is n?
2) Does the meeting with a_1 happen more than once, or infinitely often, or does the
figure continually spiral away from a_1?
Hoping someone has some software to attack this problem and maybe provide a
diagram of proceedings.
Addenda for research:
(i) What if a_1 = 1 unit, a_2 = 2 units, a_3 = 3 units, etc., with the above angles?
(ii) What if, given the same lengths in (i), but all the angles are 1º?
3 AnswersMathematics7 years agoCan you find the value of (a^2 + b)?
Two circles with a common centre have radii, r = 1 and R = 2. The annulus contains
an ellipse of maximum area, with a = semi-major axis and b = semi-minor axis.
Can you find the value of (a^2 + b) ?
3 AnswersMathematics7 years agoIn one respect, why do integer-sided scalene triangles seem to behave so well?
Q.1: What is the smallest perimeter of an integer-sided scalene triangle,
such that the quotient (circumradius / inradius) is an integer?
Q.2: Same as question 1, but the quotient is a different integer.
Q.3: (For bonus points) Same again, but quotient is yet another different integer.
Q.4: (More bonus points) Can you explain what is going on?
4 AnswersMathematics8 years agoWhat is the exact distance between these two points?
This is a spin-off from Rita the dog's recent question. See diagram at -
http://au.answers.yahoo.com/question/index;_ylt=Ah...
Place the yellow triangle, as it appears in the diagram, along the X-axis with the
left-hand vertex on the origin, (0, 0). Then position the triangles in the order, blue,
bright green, red and finally, dull green. There will be a gap between the dull green
and yellow triangles. Let point A be the 90º vertex of the dull green triangle.
Question: what is the exact distance between A and the origin?
When you have the answer, consider it in relation to the sides of the triangles, as if
they were measured in millimetres, i.e., imagine the triangles to be approximately
106 metres x 101 metres x 33 metres. Do you think the gap would be visible at the
scale of the diagram, and thus, can we regard Maple's effort as adequate?
1 AnswerMathematics8 years agoWhat's the greatest percentage coverage of a circle by 3 internal rectangles, if overlapping not allowed?
I'll be surprised if this question hasn't been asked before, but I couldn't find one.
4 AnswersMathematics8 years agoHow high is pi (or a new take on an old problem)?
On one side of a tall rectangular prism, made from a stack of 1cm cubes, a straight
line (call it L), is drawn from the lower left corner of the bottom cube to the upper right
corner of the top cube. Somewhere along this prism side, L crosses the horizontal
junction line between two cubes, such that a/b = π, where a = length of junction line
to the left of L, while b = length of junction line to the right of L.
Questions: If the bottom cube is No.1 and π is measured to within 10^(-13), then -
(a) What is the least height of the prism?
(b) Between which numbered cubes lies the particular junction?
1 AnswerMathematics8 years agoCan you solve this parabolic problem?
Pluto Planar, a resident of Flatland, slid a square, ABCD (side = 2 units),
down the right-hand arm of the parabolic well, y = x^2. It came to rest
with B and C tangent to the right arm, and D tangent to the left arm.
Question: Why did his calculation of the area below line CD, that
is bounded by the parabola, cause him to feel mildly amazed?
2 AnswersMathematics8 years agoRegarding area, is there a larger isosceles triangle in this figure?
Diagram: http://s205.beta.photobucket.com/user/falzoon/medi...
A semicircle, BCD, sits atop an isosceles triangle, ABD, with EA = 125 and
ED = 50 units, looking like a flat, tasteless version of one of my favourite foods.
Question:
Is ABD the largest isosceles triangle that can fit inside the whole figure, ABCD?
If not, then for proof of the largest one, I'll need at least one parameter, e.g.,
area, side or angle, as a decimal value, or preferably, an exact expression.
3 AnswersMathematics8 years agoAre there points A and B with the following conditions?
Are there points A on y = x^3 and points B on y = x^3 + |x| + 1,
such that the distance between A and B does not exceed 0.01?
2 AnswersMathematics8 years agoFind the ratio a/b in this folding problem?
Diagram: http://s205.beta.photobucket.com/user/falzoon/medi...
Divide a unit square (ADEH) into three equal rectangles, then fold point A onto
segment FG, such that point B lies on segment CF. Find ratio a/b (= EA / AH).
1 AnswerMathematics8 years ago
