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Arrange and write the decimals in increasing order 14.367, -28.7784, 213.22, -361.238 ,5.2 ,-5.33

I need help I am in 6th grade

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Medford Public Schools Department of Mathematics


Medford Public Schools Department of Mathematics. ... Arrange and write the decimals in increasing order. ... −361.238, −28.7784, −5.33,5.2, 14.367, 213.22 b)
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Decimal Numbers - SlideShare


Compare and order decimal numbers. 5. ... How to read and write decimals or decimal numbers? ... Actual Difference/ Estimated Difference 7. 14.255 14.000 8. 28.267 ...
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math.uga.edu


367 6 4. 2 0. 3 0 1. 4 0 1. 5 0 1. 6 0 1. 7 0 1. 8 0 1. 9 0 1. 10 0 1. 11 0 1. 12 0 1. 13 0 1. 14 0 2. 15 0 ... 5. 22 5. 23 5. 24 5. 0. 1. 2. 3. 4. 5. 8 ...
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Run_Issue_14_1985_Feb by Zetmoon - issuu


... Run_Issue_14_1985_Feb, Author: ... P0KE1 98 , 4: POKE631 ,28:POKE63 2,21 1 :POKE633,157:POKE63 4,5. 204 ... per order. Money order. Write for a Complete free Catalog,
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Calaméo - Pre Calculo


... ᎏ or ᎏ 2 5 ᎏ Use point-slope form to write the ... 0 22. 5 22. 5 22. 5 ... TV Reading 20 8. 5 32 3. 0 42 1. 0 12 4. 0 5 14. 0 28 4. 5 33 7. 0 18 12. 0 ...
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Archived Problems - Project Euler


21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95 78 17 53 28 22 75 31 67 15 94 ... 1,3,5,15 21: 1,3,7,21 28: 1,2,4,7,14,28. ... not the order. {1, 2, 3, 4 ...
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OpenSourceChemistryCourseGrades9-12 | Ion | Acid


Feb 17, 2009 · OpenSourceChemistryCourseGrades9-12 ... 28) Write the correct formula for Barium Phosphate when Ba ... Eliminate 2 213 Type of reaction?:
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31 CFR Ch. II (7-1-14 Edition) Fiscal Service, Treasury Title ...


Title 28 through Title 41 . ... call 202-741-6000 or write to the Director, ... The parts in these volumes are arranged in the following order: ...
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FRASER - St. Louis Fed - Discover Economic History


... .187 14,690,655 +i,oae,5»2 14.32l,0; ... Import«ofwheat.owt.l3,361.292 Imports of flour 2.727.642 ... 29% 3,501 22 July 31 34% Jan. • 28% ...
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Suggested Questions And Answer :


10 vedic maths rules for class IX

2 instead of 5: 34/5 can be calculated by multiplying 34 by 2 instead of dividing by 5. 34*2=68. We just move the decimal point one place to the left: 34/5=6.8. 124/5=24.8 because 124*2=248. Move the decimal point: 248 becomes 24.8. 34*5 is the same as 34/2=17 but we add a zero to make 17 into 170. 73*5 is the same as 73/2=36.5 then move the decimal point one place to the right (or add zero): 36.5 becomes 365=73*5. So we only need to know how to multiply and divide by 2 to divide and multiply by 5. We just move the decimal point. Divisibility by 9 or remainder after dividing by 9. All multiples of 9 contain digits which added together give 9. As we add the digits together, each time the result goes over 9 we add the digits of the result together and use that result and continue in this way up to the last digit. Is 12345 divisible by 9? Add the digits together 1+2+3=6. When we add 4 we get 10, so we add 1 and zero=1 then we add 5 to get 6. The number is not exactly divisible by 9, but the remainder is 6. We can also ignore any 9's in the number. Now try 67959. We can ignore the two 9's. 6+7=13, and 1+3=4; 4+5=9, so 67959 is divisible by 9. Multiplying by 11. Example: 132435*11. We write down the first and last digits 1 ... 5. Now we add the digits in pairs from the left a digit step at a time. So 1+3=4; 3+2=5: 2+4=6; 4+3=7; 3+5=8. Write these new digits between 1 and 5 and we get 1456785=132435*11. But we had no carryovers here. Now try 864753*11. Write down the first and last digits: 8 ... 3. 8+6=14, so we cross out the 8 and replace it with 8+1=9, giving us 94 ... 3. Next pair: 6+4=10. Again we go over 10 so we cross out 4 and make it 5. Now we have 950 ... 3. 4+7=11, so we have 9511 ... 3. 7+5=12, giving us 95122 ... 3; 5+3=8, giving us the final result 9512283.  Divisibility by 11. We add alternate digits and then we add the digits we missed. Subtract one sum from the other and if the result is zero the original number was divisible by 11. Example: 1456785. 1 5 7 5 make up one set of alternate digits and the other set is 4 6 8. 1+5+7=13. We drop the ten and keep 3 in mind to add to 5 to give us 8. Now 4 6 8: 4+6=10, drop the ten and add 0 to 8 to give us 8 (or ignore the zero). 8-8=0 so 11 divides into 1456785. Now 9512283: set 1 is 9 1 2 3 and set 2 is 5 2 8; 9+1=0 (when we drop the ten); 2+3=5; set 1 result is 5; 5+2+8=5 after dropping the ten, and 5-5=0 so 9512283 is divisible by 11. Nines remainder for checking arithmetic. We can check the result of addition, subtraction, multiplication and (carefully) division. Using Method 2 above we can reduce operands to a single digit. Take the following piece of arithmetic: 17*56-19*45+27*84. We'll assume we have carried out this sum and arrived at an answer 2365. We reduce each number to a single digit using Method 2: 8*2-1*9+9*3. 9's have no effect so we can replace 9's by 0's: 8*2 is all that remains. 8*2=16 and 1+6=7. This tells us that the result must reduce to 7 when we apply Method 2: 2+3+6=11; 1+1=2 and 2+5=7. So, although we can't be sure we have the right answer we certainly don't have the wrong answer because we arrived at the number 7 for the operands and the result. For division we simply use the fact that a/b=c+r where c is the quotient and r is the remainder. We can write this as a=b*c+r and then apply Method 2, as long as we have an actual remainder and not a decimal or fraction. Divisibility by 3. This is similar to Method 2. We reduce a number to a single digit. If this digit is 3, 6 or 9 (in other words, divisible by 3) then the whole number is divisible by 3. Divisibility by 6. This is similar to Method 6 but we also need the last digit of the original number to be even (0, 2, 4, 6 or 8). Divisibility by 4. If 4 divides into the last two digits of a number then the whole number is divisible by 4. Using 4 or 2 times 2 instead of 25 for multiplication and division. 469/25=469*4/100=1876/100=18.76. 538*25=538*100/4=134.5*100=13450. We could also double twice: 469*2=938, 938*2=1876, then divide by 100 (shift the decimal point two places to the left). And we can divide by 2 twice: 538/2=269, 269/2=134.5 then multiply by 100 (shift the decimal point two places left or add zeroes). Divisibility by 8. If 8 divides into the last three digits of a number then the whole number is divisible by 8. Using 8 or 2 times 2 times 2 instead of 125 for multiplication and division. Similar to Method 9, using 125=1000/8. Using addition instead of subtraction. 457-178. Complement 178: 821 and add: 457+821=1278, now reduce the thousands digit by 1 and add it to the units: 278+1=279; 457-178=279. Example: 1792-897. First match the length of 897 to 1792 be prefixing a zero: 0897; complement this: 9102. 1792+9102=1894. Reduce the thousands digit by 1 and add to the result: 894+1=895. Example: 14703-2849. 2849 becomes 02849, then complements to 97150. 14703+97150=111853; reduce the ten-thousands digit by 1 and and add to the result: 11854. Squaring numbers ending in 5. Example: 75^2. Start by writing the last two digits, which are always 25. Take the 7 and multiply by 1 more than 7, which is 8, so we get 56. Place this before the 25: 5625 is the square of 75. The square of 25 is ...25, preceded by 2*3=6, so we get 625. All numbers ending in 0 or 5 are exactly divisible by 5 (see also Method 1). All numbers ending in zero are exactly divisible by 10. All numbers ending in 00, 25, 50 or 75 are divisible by 25. Divisibility by 7. Example: is 2401 divisible by 7? Starting from the left with a pair of digits we multiply the first digit by 3 and add the second to it: 24: 3*2+4=10; now we repeat the process because we have 2 digits: 3*1+0=3. We take this single digit and the one following 24, which is a zero: 3*3+0=9. When we get a single digit 7, 8 or 9 we simply subtract 7 from it: in this case we had 9 so 9-7=2 and the single digit is now 2. Finally in this example we bring in the last digit: 3*2+1=7, but 7 is reduced to 0. This tells us the remainder after dividing 2401 by 7 is zero, so 2401 is divisible by 7. Another example: 1378. 3*1+3=6; 3*6=18 before adding the next digit, 7 (we can reduce this to a single digit first): 3*1+8=3*1+1=4; now add the 7: 4+7=4+0=4;  3*4=12; 3*1+2+8=5+1=6, so 6 is the remainder after dividing 1378 by 7.  See also my solution to: http://www.mathhomeworkanswers.org/72132/addition-using-vedic-maths?show=72132#q72132
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solve by factorisation the following equation (1).5x^2+2x-15=0 (2).21x^2-25x=4 (3).10x^2+3x-4=0

  (1) 5x^2+2x-15=0 I suspect that this should be 5x^2+22x-15=0, because the equation as written does not factorise. Write down the factors of the squared term coefficient and the constant term. We write these as ordered pairs (a,b) squared term (c,d): (1,5) and constant: (1,15), (15,1), (3,5), (5,3). The sign of the constant (minus) tells us that we are going to subtract the cross-products of  factors. If it had been plus we would be adding the cross-products. Now we create a little table: Quadratic factor table a b c d ad bc  |ad-bc| 1 5 15 1 1 75 74 1 5 1 15 15 5 10 1 5 3 5 5 15 10 1 5 5 3 3 25 22 The column |ad-bc| just means the difference between ad and bc, regardless of it being positive or negative, just take the smaller product away from the larger. If the coefficient of the x term is in the last column, then that row contains the factors you need. If it isn't in the last column, then the quadratic doesn't factorise, or you've missed some factors. If 22x is the middle term, then the factors are shown in the last row and (a,b,c,d)=(1,5,5,3). Now we look at the sign of the middle term. Whatever the sign is we put it in front of the number c or d for the larger cross-product. The cross-products are bc and ad. In this case, bc is bigger than ad so the + sign goes in front of c. The sign in front of the constant tells us whether the sign in front of d is different or the same. If the sign is plus it's the same, otherwise it's the opposite sign. So in this case, it's minus, so the minus sign goes in front of d and we have (ax+c)(bx-d) (note the order of the letters!) or (x+5)(5x-3), putting in the values for a, b, c and d. If (x+5)(5x-3)=0, then x+5=0 or 5x-3=0. So in the first case x=-5 and in the second case 5x=3 so x=3/5. (2) 21x^2-25x-4=0 (moving 4 over to the left to put the equation into standard form). Quadratic factor table a b c d ad bc  |ad-bc| 1 21 1 4 4 21 17 1 21 4 1 1 84 83 1 21 2 2 2 42 40 3 7 1 4 12 7 5 3 7 4 1 3 28 25 3 7 2 2 6 14 8 The row in bold print applies. The sign in front of 4 is minus so the signs in the brackets will be different. 28 is the larger product, so since we have -25x, the minus sign goes in front of c (4) and plus in front of d (1): (3x-4)(7x+1)=0. So the solution is x=4/3 or -1/7. (3) 10x^2+3x-4=0. Quadratic factor table a b c d ad bc  |ad-bc| 1 10 1 4 4 10 6 1 10 4 1 1 40 39 1 10 2 2 2 20 18 2 5 1 4 8 5 3 2 5 4 1 2 20 18 2 5 2 2 4 10 6 (2x-1)(5x+4)=0, so x=1/2 or -4/5.  
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write an equation that can be used to medel the data in table.

An equation that models the data is: y=(x^3)/2-2x^2+9x/2+1. To find the equation I used the fact that there are 4 data pairs and I can find 4 unknowns from 4 equations. I also know that if the ordered pair is (x,y), y increases with x, and y is defined for x in {0, 1, 2, 3}. Let y=ax^3+bx^2+cx+d. This equation has 4 unknown coefficients a to d. When x=0, y=1 so d=1. When x=1, y=4 so a+b+c+d=4 and a+b+c=3. c=3-(a+b). When x=2, y=6 so 8a+4b+2c+d=6 and 8a+4b+2(3-(a+b))=5. 6a+2b=-1, and b=-(6a+1)/2. Therefore c=3-(a-(6a+1)/2)=(4a+7)/2. When x=3, y=10 so 27a+9b+3c+1=10 and 27a-9(6a+1)/2+3(4a+7)/2=9. 54a-9-54a+12a+21=18 and 12a=6, so a=1/2. From a=1/2 we get b=-2 and c=9/2. If we substitute the data pairs we find that they satisfy the equation y=(x^3)/2-2x^2+9x/2+1. Therefore this equation models the data.
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How many different ways can Hector put 5 tulips in 2 vases?

Ordered pair (V1,V2) represents the number of tulips in the two vases. V1 and V2 identify the vases. The set of possibilities are: (5,0), (4,1). (3,2), (2,3), (1,4), (0,5) if the tulips are identical. If the tulips are distinct: A, B, C, D, E, the permutations are: (5,0): (ABCDE,0) [1] (4.1): (BCDE,A), (ACDE,B), (ABDE,C), (ABCE,D), (ABCD,E) [5] (3,2): (ABC,DE), (ABD,CE), (ABE,CD), (ACD,BE), (ACE,BD), (ADE,BC), (BCD,AE), (BCE,AD), (BDE,AC), (CDE,AB) [10] (2,3): (DE,ABC), (CE,ABD), (CD,ABE), (BE,ACD), (BD,ACE), (BC,ADE), (AE,BCD), (AD,BCE), (AC,BDE), (AB,CDE) [10] (1,4): (A,BCDE), (B,ACDE), (C,ABDE), (D,ABCE), (E,ABCD) [5] (0,5): (0,ABCDE) [1] [32 ways in total=1+5+10+10+5+1]  If the tulips in each vase can be placed in different orders, then there are more permutations: 2 ways of arranging 2 tulips; 6 ways of arranging 3; 24 ways for 4; 120 ways for 5. For example, for (BC,ADE) there are 2 ways to arrange BC, and 6 ways to arrange ADE. So (BC,ADE) spawn 2*6=12 arrangements. Since there are 10 (2,3) arrangements, each of these spawns 12, giving a total of 120 possible ways. For (1,4) each spawns 24, making 5*24=120 possible ways. For (0,5) again we have 120 ways. Since there are 6 rows above the grand total would be 6*120=720 ways.  
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If 1=3,2=5,3=6,4=9 so how much word needed for equal of 5?

5=10 using the following logic. Switch to the binary system of counting: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 represent the numbers 1 to 10 decimal in binary. If we cross out those numbers with an odd number of ones we get: 11, 101, 110, 1001, 1010. When these are converted back to decimal we get: 3, 5, 6, 9, 10. These are the listed numbers in order: so 1=3, 2=5, 3=6, 4=9 and 5=10. Another way of solving the problem is to list the numbers in order that are the sum of an even number of powers of 2; or the sum of a pair of powers of 2. (1) 3=2+2^0; (2) 5=2^2+2^0; (3) 6=2^2+2^1; (4) 9=2^3+2^0; (5) 10=2^3+2^1; (6) 12=2^3+2^2, etc.
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(1, -20), (7, -20), (2, -40), (6, -40), (5, -50), (8, 10), (9, 46), (0, 7), (-1, 47), (-2, 89)

First put all the points (x,y) in order of the x value: -2 89 -1 47 0 7 1 -20 2 -40 5 -50 6 -40 7 -20 8 10 9 46   When x=0 y=7, so the constant term in the function is 7 (y intercept). There's a minimum (turning point) near x=5 because the values of y for x=2 and 6 (-40) are more positive than -50. The gradient (dy/dx) gets positively steeper as x increases after x=5, and negatively steeper between x=-2 and +2. The degree of the polynomial is even because the function is positive for larger magnitude positive and negative values of x. The zeroes are between x=0 and 1 and between x=7 and 8. The function can be split into two parts: (1) y for -2 Read More: ...

how do you work out: 3+7=....(finite 5)

Not sure what you're asking here. Are you trying to sum 3 and 7 to a different base? In decimal the answer of course is 10. 10 means the base as a number to that base. If the base was 5, 10 would be the number 5 itself. If the base was 2, 10 would be the number 2. In base 5 we would count 0 1 2 3 4 10 11 12 13 14 20, corresponding to the numbers 0 to 10 in decimal. In base 5 we only use digits 0 to 4; in binary (base 2) we only have digits 0 and 1. In base 10 (decimal) we only have digits 0 to 9. In base 16 (hexadecimal) we have to invent symbols for digits we don't have: there's no single symbol after 9, so we use letters A to F to represent what we would write in decimal as 10 to 15. 10 in base 16 is the same as 16 in base 10. What's 16 in base 5? Well it's 3 times the base plus 1: 31. What is 16 in binary (base 2)? 10 stands for 2; 100 stands for 4 (2 squared); 1000 stands for 8 (2 cubed); 10000 stands for 16 (2^4). 3+7=20 in base 5 and the sum totally in base 5 would be 3+12=20.
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Eight horses are entered in a race...(Have more)

1(a) How many different orders are possible for completing the race? 8*7*6*5*4*3*2*1=8!=40,320 (b) In how many different ways can first, second, and third places be decided? (Assume there is no tie.) (8*7*6)=P(8,3)=336 or 6*C(8,3) where 6=3*2*1 the number of ways of arranging three items 2.)Telephone numbers consist of seven digits; the first digit cannot be 0 or 1. How many telephone numbers are possible? 8*10^6=8,000,000  3.)In how many ways can five people be seated in a row of five seats? 5*4*3*2*1=5!=120 4.)In how many ways can five different mathematics books be placed next to each other on a shelf? 5!=120 5.)In a family of four children, how many different boy-girl birth-order combinations are possible? (The birth orders BBBG and BBGB are different.) 16=2*2*2*2 from BBBB to GGGG 6.)Two cards are chosen in order from a deck. In how many ways can this be done if (a) the first card must be a spade and the second must be a heart?  13*13=169 (b) both cards must be spades? 13*12=156 7.)A company’s employee ID number system consists of one letter followed by three digits. How many different ID numbers are possible with this system? 26*10*10*10=26,000 Continued in comment...
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x+y+z=9,2x-3y+4z=13,2x-3y+4z=13,3x+4y+5z=40. solve

There are 4 equations but only 3 unknowns. This means one equation is unnecessary or there is inconsistency. Call the equations in order A, B, C and D. B and C are the same so we can remove one. Let's remove C. 3A+B: 3x+3y+3z+2x-3y+4z=27+13; 5x+7z=40. So 5x=40-7z. We have x in terms of z. D-3A: y+2z=40-27=13. So y=13-2z. We have y in terms of z. We can substitute for x and y in one equation (choose A) to leave z as the only variable: (40-7z)/5+13-2z+z=9. Multiply through by 5: 40-7z+65-10z+5z=45; -12z+60=0, so 12z=60 and z=5. We can now find x and y: 5x=40-7z=40-35=5, making x=1. y=13-2z=13-10=3. So the solution is x=1, y=3 and z=5. Substitute these values in the original equations to check them out.   Gauss-Jordan method: Write the equations in matrix format: [ 1 1 1 | 9 ] [ 2 -3 4 | 13 ] [ 3 4 5 | 40 ] R2→R2-2R1 [ 1 1 1 | 9 ] [ 0 -5 2 | -5] [ 3 4 5 | 40 ] R3→R3-3R1: [ 1 1 1 | 9 ] [ 0 -5 2 | -5 ] [ 0 1 2 | 13 ] R3→5R3+R2: [ 1 1 1 | 9 ] [ 0 -5 2 | -5 ] [ 0 0 12 | 60 ] R2→-R2+R3 then R2→(1/5)R2 and R3→(1/12)R3: [ 1 1 1 | 9 ] [ 0 1 2 | 13 ] [ 0 0 1 | 5 ] R2→R2-R3: [ 1 1 1 | 9 ] [ 0 1 1 | 8 ] [ 0 0 1 | 5 ] R1→R1-R2: [ 1 0 0 | 1 ] [ 0 1 1 | 8 ] [ 0 0 1 | 5 ] R2→R2-R3: [ 1 0 0 | 1 ] [ 0 1 0 | 3 ] [ 0 0 1 | 5 ] From this identity matrix x=1, y=3 and z=5.  
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What is the diameter of a spiral coil of .65265 inch diameter pipe 100 feet long?

The equation of a spiral in polar coordinates has the general form r=A+Bø, where A is the starting radius of the spiral and B is a factor governing the growth of the spiral outwards. For example, if B=0, there is no outward growth and we just have a circle of radius A. A horizontal line length A represents the initial r, and the angle ø is the angle between r and this horizontal line. So r increases in length as ø increases (this angle is measured in radians where 2(pi) radians = 360 degrees, so 1 radian is 180/(pi)=57.3 degrees approximately.) If B=1/2 and A=5", for example, the minimum radius would be 5" when ø=0. When ø=2(pi) (360 degrees), r=5+(pi), or about 8.14". This angle would bring r back to the horizontal position, but it would be 8.14" instead of the initial 5". At ø=720 degrees, the horizontal line would increase by a further 3.14". Everywhere on the spiral the spiral arms would be 3.14" apart. What would B be if the spiral arms were 0.65625" apart? 2(pi)B=0.65625, so B=0.65625/(2(pi))=0.10445". The equation of the spiral is r=5+0.10445ø. To calculate the length of the spiral we have two possible ways: an approximate value based on the similarity between concentric circles and a spiral; or an accurate value obtainable through calculus. The approximate way is to add together the circumferences of the concentric circles: L=2(pi)(5+(5+0.65625)+...+(5+0.65625N)) where L=spiral length and N is the number of turns. L=2(pi)(5N+0.65625S) where S=0+1+2+3+...+(N-1)=N(N-1)/2. This formula arises from the fact that the first and last terms (0, N-1) the second and penultimate terms (1, N-2) and so on add up to N-1. So, for example, if N were 10 we would have (0+9)+(1+8)+(2+7)+(3+6)+(4+5)=5*9=45=10*9/2. If N were 5 we would have 0+1+2+3+4=10=(0+4)+(1+3)+2=5*4/2. L=12*100 inches. L=1200=2(pi)(5N+0.65625N(N-1)/2)=(pi)N(10+0.65625(N-1))=(pi)N(9.34375+0.65625N). If the external radius is r1 and the internal radius is r then the thickness of the spiral is r1-r and since 0.65625 is the gap between the spiral arms N=(r1-r)/0.65625. N is an integer, but, since it is unlikely that this equation would actually produce an integer we would settle for the nearest integer. If we solve this equation for N, we can deduce the external radius and diameter of the spiral: N(9.34375+0.65625N)=1200/(pi)=381.97; 0.65625N^2+9.34375N-381.97=0 and N=(-9.34375+sqrt(1089.98))/1.3125=18 (nearest integer). This means that there are 18 turns of the spiral to make the total length about 100 feet. If X is the final external diameter of the coiled pipe and the internal radius is 5" (the minimum allowable) then X/2 is the external radius, so N=((X/2)-5)/0.65625. We found N=18 so we can find X: X=2*(0.65625*18+5)=33.625in. Solution using calculus Using calculus, we can work out the relationship between the length of the spiral and other parameters. We start with any polar equation r(ø) and a picture: draw a line representing a general value of r. At a small angle dø to this line we draw another line a little bit longer, length r+dr. Now we join the ends together to make a narrow-angled triangle AOB where angle AOB=dø and AB=ds, the small section of the curve. In the triangle AO is length r and BO is length r+dr. If we mark the point C along BO so that CO is length r, the same as AO, we have an isosceles triangle COA. Because the apex angle is small, CA=rdø, the length of the arc of the sector. In triangle ABC, CB=dr, AB=ds and CA=rdø. By Pythagoras, AB^2=CB^2+CA^2, that is, ds^2=dr^2+r^2dø^2, because angle BCA is a right angle as dø tends to zero. The length of the curve is the result of adding the tiny ds values together between limits of r or ø. We can write ds=sqrt(dr^2+r^2dø^2). If we divide both sides by dr, we get ds/dr=sqrt(1+(rdø/dr)^2) so s=integral(sqrt(1+(rdø/dr)^2)dr, where s is the length of the curve. The integral is definite if we define the limits of r. For our spiral we have r=A+Bø, making ø=(r-A)/B and B=p/(2(pi)), where p is the diameter of the pipe=0.65625", so we can substitute for ø in the integral and the limits for r are A to X/2, where A is the inner radius (A=5") and X/2 is the outer radius. dø/dr=2(pi)/p, a constant=9.57 approx. s=integral(sqrt(1+(2(pi)r/p)^2)dr) between limits r=A to X/2. After the integral is calculated, we solve for X putting s=1200". The expression (2(pi)r/p)^2 is large compared to 1, so s=integral((2(pi)r/p)dr) approximately and s=[(pi)r^2/p] (r=A to X/2); therefore, since we know s=1200, we can write ((pi)/p)(X^2/4-A^2)=1200. Therefore X=2sqrt(1200p/(pi))+A^2)=33.21". Compare this answer with the one we got before and we can see they are close. [We could get a formal solution to the integral, using hyperbolic trigonometric or other logarithmic functions, but such a solution would make it very difficult or tedious to solve for X, since X would appear in logarithmic expressions and in other expressions making it difficult or impossible to isolate X. For example, the next term in the expansion of the integral would be (p/(4(pi))ln(X/2A), having a value of about 0.06. It is anticipated, therefore, that an approximation would be sufficient in this problem with the given figures.] We can feel justified in using the formula for finding the length of pipe, L, when X=6'=72": L=((pi)/p)(1296-25)=6084.52"=507' approximately. This length of pipe would hold 507/100*0.96 gallons=4.87 gallons.      
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