Quizzes & Puzzles1 min ago
Algebra
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Here’s a question for you mathematically minded readers
I want to build some steps out of blocks to get to the top of a platform 250 feet high
I want the top of the steps to be level with the top of the platform or just below
I can order a total of 1,126,576 linear feet of block material which I intend to cut up into one foot cubes
if I build steps with these with uniform rise how many columns will there be, and what will the rise be using all the material.
I want to build some steps out of blocks to get to the top of a platform 250 feet high
I want the top of the steps to be level with the top of the platform or just below
I can order a total of 1,126,576 linear feet of block material which I intend to cut up into one foot cubes
if I build steps with these with uniform rise how many columns will there be, and what will the rise be using all the material.
Answers
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There is a story about the mathematician Gauss who when he was a schoolboy, his teacher needed to go out asked the class to add up the first 1000 numbers while he was out - before his hand was on the door Gauss gave him the answer.
He pictured it as a square cut in half and came up with the formula (N+1)(N/2)
in your case we want to do it backwards
1,126,576=(N+1)(N/2)
½N²+½N-1,126,576 =0
solving this as a quadratic gives us a number a bit over 1500
so we have 1500 steps each rising 1 foot above the other so you've 1500 feet
If I've understood what you're asking right
In terms of a 250 foot platform you may have over ordered
But linear feet!! is there a prize in groats?
He pictured it as a square cut in half and came up with the formula (N+1)(N/2)
in your case we want to do it backwards
1,126,576=(N+1)(N/2)
½N²+½N-1,126,576 =0
solving this as a quadratic gives us a number a bit over 1500
so we have 1500 steps each rising 1 foot above the other so you've 1500 feet
If I've understood what you're asking right
In terms of a 250 foot platform you may have over ordered
But linear feet!! is there a prize in groats?
Do you want to know how wide the steps can be?
To build steps 1 foot wide you need 1 + 2 + 3 + .... + 248 + 249 + 250 blocks. To calcul;ate the total, we pair these numbers, taking one from the beginning and the other fom the end:
1 + 250 = 251
2 + 249 = 251
3 + 248 = 251
and so on until
123 + 124 = 251
124 + 127 = 251
125 + 126 = 251
So we now have 125 "paired" numbers, each equalling 251.
So the total number of blocks, to build steps 250 feet high is 251 x 125 = 15750. (This is how Jake's formula (N+1)(N/2) is derived).
With 1,126,576 blocks, you can build a stairway 1,126,576 / 15750 = 71 blocks wide (with some about 8,00 spare blocks!)
(Before anyone corrects me,I know that my method would need a tweak to deal with an odd number of steps)
To build steps 1 foot wide you need 1 + 2 + 3 + .... + 248 + 249 + 250 blocks. To calcul;ate the total, we pair these numbers, taking one from the beginning and the other fom the end:
1 + 250 = 251
2 + 249 = 251
3 + 248 = 251
and so on until
123 + 124 = 251
124 + 127 = 251
125 + 126 = 251
So we now have 125 "paired" numbers, each equalling 251.
So the total number of blocks, to build steps 250 feet high is 251 x 125 = 15750. (This is how Jake's formula (N+1)(N/2) is derived).
With 1,126,576 blocks, you can build a stairway 1,126,576 / 15750 = 71 blocks wide (with some about 8,00 spare blocks!)
(Before anyone corrects me,I know that my method would need a tweak to deal with an odd number of steps)
Made a complete pig's of this, ignore above, I'll start agian.
Do you want to know how wide the steps can be?
To build steps 1 foot wide you need 1 + 2 + 3 + .... + 248 + 249 + 250 blocks. To calculate the total, we pair these numbers, taking one from the beginning and the other fom the end:
1 + 250 = 251
2 + 249 = 251
3 + 248 = 251
and so on until
123 + 124 = 251
124 + 127 = 251
125 + 126 = 251
So we now have 125 "paired" numbers, each equalling 251.
So the total number of blocks, to build steps 250 feet high is 125 x 251 = 31,375. (This is how Jake's formula (N+1)(N/2) is derived).
With 1,126,576 blocks, you can build a stairway 1,126,576 / 15750 = 35 blocks wide (with some about 30,000 spare blocks!)
(Before anyone corrects me,I know that my method would need a tweak to deal with an odd number of steps)
Do you want to know how wide the steps can be?
To build steps 1 foot wide you need 1 + 2 + 3 + .... + 248 + 249 + 250 blocks. To calculate the total, we pair these numbers, taking one from the beginning and the other fom the end:
1 + 250 = 251
2 + 249 = 251
3 + 248 = 251
and so on until
123 + 124 = 251
124 + 127 = 251
125 + 126 = 251
So we now have 125 "paired" numbers, each equalling 251.
So the total number of blocks, to build steps 250 feet high is 125 x 251 = 31,375. (This is how Jake's formula (N+1)(N/2) is derived).
With 1,126,576 blocks, you can build a stairway 1,126,576 / 15750 = 35 blocks wide (with some about 30,000 spare blocks!)
(Before anyone corrects me,I know that my method would need a tweak to deal with an odd number of steps)
There are a lot of "impracticalities" on this question ! (not least the > 1 million sawing up into 1ft. blocks... good luck with that....)
But seriously , the large excess of material left over leads me to think that they are looking for a 3D structure . such as half of a square-based pyramid leading up to the apex 250 or 249 feet high (the last 1 ft being the step up to the platform ) . This would be more stable than a 250ft . steps only 1ft wide .
I'm not sure how to calculate the number of blocks that this would use.
But seriously , the large excess of material left over leads me to think that they are looking for a 3D structure . such as half of a square-based pyramid leading up to the apex 250 or 249 feet high (the last 1 ft being the step up to the platform ) . This would be more stable than a 250ft . steps only 1ft wide .
I'm not sure how to calculate the number of blocks that this would use.
I also worked this out this morning using the n(n+1)/2 formula that results in 31,375 foot cube blocks needed for 250 steps 1 foot wide. After that it depends on how wide you want your steps and the dimensions of your original - what is the depth and height of your linear foot of material? In reality 250 steps a foot high at a time would need Rocky 111 to climb them without suffering a seizure.
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