Design Four BASIC Walls

[Otis]

A discredited man wanted to be king, so he schemed to convince people that he was great again. Toward this end, he commissioned four great backdrops for his public appearances.

"They gotta look like golden walls rising from a dark horizon", he demanded of his apprentices. One crony replied, "I know where we can get some leftover, flashy gold paint for free."

"Glittery and cheap?", mused the would-be king. "Perfect!"

A representative template of one such backdrop is attached below. The "wall" is QRST.

Q is located at the left end of a semi-circle's base. In the center of this base, a vertical support pole is set at T, and a taut guy wire runs from R to an anchor at A such that the wire is tangent to the semi-circle at S.

There is enough paint to cover 4,617 square meters of the lowest-quality plywood available. Due to stability concerns, the semi-circle radius cannot exceed 50 meters in length, and aesthetics dictate that neither the height of the support pole nor the distance from T to A can exceed twice the radius.

Find any four variations of this design, and state the radius and distance from T to A for each, such that the combined area of all four "walls" is exactly 4,617 square meters.

 

[Otis]

Issue resolved. Image posting not working like it used to.

 

[Otis]

Hint: If the paint were applied one "stroke" at a time, with each "stroke" covering exactly 3/10ths of a square meter, then the number of strokes for each wall would be an Integer.

 

[Otis]

sum of areas of right triangles RST and QRT = integer

Not necessarily.

The number of strokes to paint that sum of areas is an integer.

4 different cases of above have total area = 4617

Yes?

Yes.

Or you mean same applies to each of the 4 cases;
each / (3/10) = integer

Yes (could have been better worded) -- because no stoke can span two different walls. The monstrosities are built first and gussied up afterwards.

 

[Otis]

If area=4618 was the target:
(radius,pole,area)
1: 20,25,400
2: 424,26,432
3: 48,52,1728
4: 42,58,2058

Correct?

I don't know; the only aggregate area that I considered was 4,617 square meters.

PS: Puzzle calls for distance from T to A; kindly check your "pole".

 

[Otis]

Solution submissions must be formatted per instructions. Kindly note the emphasis below.

Find any four variations of this design, and state the radius and distance from T to A for each ...

 

[Otis]

Wall1: r = 3, TA = 3.75
Wall2: r = 6, TA = 7.5
Wall3: r = 27, TA = 36
Wall4: r = 48, TA = 60

That must be a typo, yes?

So I believe that FULLY answers your question.

More than fully, as it only asked for one solution. Well done!

(By the way, there are more solutions than you think.)

Care to kick it up a notch?

Same puzzle with same hint, but include this extra info: For any "wall" whose area is not a whole number of square meters, the number of strokes required to paint the fractional part of the area is also an integer.

 

[Otis]

I'll pass...

Fair 'nuff.

Cats don't like coding; their paws are too big.

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r = 49, TA = 92.75

r = 36, TA = 54.75

r = 24, TA = 36.5

r = 21, TA = 39.75

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