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# Recent problems solved by 'AnlytcPhil'

AnlytcPhil answered: 1274 problems
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 Sequences-and-series/552544: I need some help with some homework. I'm struggling with these four problems: 1. What is the twenty-fifth term of the arithmetic sequence with a1 = 1 and d = 10? 2.What is the twenty-ninth term of arithmetic sequence with a1 = 13 and d = 5/2? 3. What are the two arithmetic means between 13 and 8? 4. What is Sn for the arithmetic series with d = 4, an = 27, and n = 9? Thanks so much!1 solutions Answer 360394 by AnlytcPhil(1277)   on 2012-01-04 11:51:42 (Show Source): You can put this solution on YOUR website!```1. What is the twenty-fifth term of the arithmetic sequence with a1 = 1 and d = 10? Substitute a1=-1, d=-10, and n=25 in an = a1 + (n-1)d Answer: -241 2.What is the twenty-ninth term of arithmetic sequence with a1 = 13 and d = 5/2? Substitute a1=13, d=-5/2, and n=29 in an = a1 + (n-1)d Answer: -57 3. What are the two arithmetic means between 13 and 8? a1 - -13, a2 = ?, a3 = ?, a4 = 8 Each term differs from the preceding term by d. a2 is a1 + d, and a3 is a4 - d, so a1 = -13, a2 = -13+d, a3 = 8-d, a4 = 8 a2 + d = a3 -13+d+d = 8-d -13+2d = 8-d 3d = 21 d = 7 a2 = -13+7 = -6 a3 = 8-7 = 1 -13, -6, 1, 8 They are -6 and 1 4. What is Sn for the arithmetic series with d = 4, an = 27, and n = 9? First substitute an = 27, d=-4, and n=9 in an = a1 + (n-1)d and solve for a1. Get a1 = 59 Substitute a1 = 59, d = -4, leave n as just n in Sn = (n/2)[a1 + (n-1)d] Simplify Get Sn = n(61-2n) Edwin```
 Polynomials-and-rational-expressions/552552: Divide: 1 solutions Answer 360389 by AnlytcPhil(1277)   on 2012-01-04 11:17:57 (Show Source): You can put this solution on YOUR website!``` x² - 3x + 2 x² + 3x - 18)x⁴+ 0x³ - 25x² + 62x - 36 x⁴+ 3x³ - 18x² -3x³ - 7x² + 62x -3x³ - 9x² + 54x 2x² + 8x - 36 2x² + 6x - 36 2x + 0 Answer: x²-3x+2+ Edwin```
 Numeric_Fractions/552551: Short form of 1/2-2/3+3/4-4/5+....?1 solutions Answer 360384 by AnlytcPhil(1277)   on 2012-01-04 10:54:01 (Show Source): You can put this solution on YOUR website!, a divergent series
 Quadratic-relations-and-conic-sections/552378: Can you please show me the graph of 9x^2+4y^2-54x+16y+61=0 and the coordinates of the center and foci please!1 solutions Answer 360267 by AnlytcPhil(1277)   on 2012-01-03 22:13:17 (Show Source): You can put this solution on YOUR website!``` We have to get it either in the form: + = 1 in which the ellipse will look like this "" or this form: + = 1 in which the ellipse will look like this "" 9x² + 4y² - 54x + 16y + 61 = 0 Get the x terms together, and the y terms together. 9x² - 54x + 4y² + 16y + 61 = 0 Get the 61 off the left side by adding -61 to both sides 9x² - 54x + 4y² + 16y = -61 Factor the 9 out of the first two terms on the left (factor out just the 9, not the x) 9(x² - 6x) + 4y² + 16y = -61 Factor the 4 out of the last two terms on the left (factor out just the 4, not the y) 9(x² - 6x) + 4(y² + 4y) = -61 Complete the square inside the first parentheses: Multiply the coefficient of x, which is -6 by , getting -3 Square -3 getting (-3)² = 9 Add 9 inside at the end of the first parentheses: 9(x² - 6x + 9) + 4(y² + 4y) = -61 + 81 Notice that to offset adding the 9 inside the parentheses, we had to add 81 to the right side. That's because when we added 9 inside the first parentheses on the left, that actually amounted to adding 9·9 or 81 to the left side because of the 9 coefficient in front of the first parentheses on the left. Complete the square inside the second parentheses: Multiply the coefficient of y, which is 4 by , getting 2 Square 2 getting (2)² = 4 Add 4 inside at the end of the second parentheses: 9(x² - 6x + 9) + 4(y² + 4y + 4) = -61 + 81 + 16 Notice that to offset adding the 4 inside the parentheses, we had to add 16 to the right side. That's because when we added 4 inside the second parentheses on the left, that actually amounted to adding 4·4 or 16 to the left side because of the 4 coefficient in front of the second parentheses on the left. Next we factor the two parentheses and combine the terms on the right: 9(x - 3)(x - 3) + 4(y + 2)(y + 2) = 36 Those factorizations can be written shorter as the squares of binomials: 9(x - 3)² + 4(y + 2)² = 36 In fact most people skip the step before the last one. Next we must get a 1 on the right side. So we divide each term by 36: + = We simplify: + = 1 Now we compare that to + = 1 and the ellipse will look like this "" We can tell that because a² is larger than b². On comparing the two we see that h=3, k=-2, b²=4 or b=2, a²=9, or a=3 The center is (h,k) = (3,-2). We plot the center (3,-2) We draw the major axis vertically, which is bisected at the center. We count a=3 units up from the center and a=3 units down from the center to the vertices. The vertex which is 3 units above the center (3,-2) is the point (3,1). The vertex which is 3 units down from the center is the point (3,-5): We draw the minor axis vertically, which is also bisected at the center. We count b=2 units left from the center and b=2 units right from the center to the covertices. The covertex which is 2 units left of the center (3,-2) is the point (1,-2). The covertex which is 2 units right of the center is the point (5,-2): Now we can sketch in the ellipse: All we need to do now is to find the foci. They are two points inside the ellipse on the major axis each of which is "c" units from the center, where c is calculated from this Pythagorean relation: c² = a² - b² c² = 9 - 4 c² = 5 c = . To find the coordinates of the upper focus, we add to the y coordinate of the center and get the point (3,-2+). To get the lower focus, we subtract from the y coordinate of the center and get the point (3,-2-). We plot them: Center: (3,-2), Vertices: (3,1) and (3,-5), Covertices: (1,-2) and (5,-2) Foci: (3,-2-) and (3,-2+), Major axis: 2a = 6, Minor axis: 2b = 4. Eccentricity: = Edwin```
 Exponents/552373: write in exponential notation x·x·x·x·y·y·y·z1 solutions Answer 360243 by AnlytcPhil(1277)   on 2012-01-03 20:57:43 (Show Source): You can put this solution on YOUR website!x·x·x·x·y·y·y·z = x4y3z
 Numeric_Fractions/552325: what is 9 5/9 - 6 5/6?1 solutions Answer 360220 by AnlytcPhil(1277)   on 2012-01-03 19:56:22 (Show Source): You can put this solution on YOUR website!``` We can't subtract because they don't have the same denominator. The LCD of 9 and 6 is 18, because both will divide evenly into 18. So we change to and to We still can't subtract because the upper fraction has a smaller numerator than the lower fraction . So we borrow 1 from the 9, leaving it with 8, and that gives us to add to the fraction which just amounts to adding the denominator 18 to the numerator 10, getting 28, so we have this: And now we can subtract because the numerator above is bigger than the numerator below. So 8 minus 6 gives 2 for the whole part of the answer, and 28-15 gives 13 for the numerator of the answer so we subtract and get: Edwin```
 Graphs/552296: Hi tutor, I need help on this problem. It is asking me to graph the these lines and highlight the positive slope lines pink and the negative blue. So how would you graph y=3x+5 or y=0.5x+3. Please help me with this because it is due tomorrow. thanks1 solutions Answer 360214 by AnlytcPhil(1277)   on 2012-01-03 19:22:28 (Show Source): You can put this solution on YOUR website!``` y = 3x + 5 The slope is the coefficient of x, which is 3 when the equation of the line is solved for y. It is positive so I colored it pink Here is the line: y = 0.5x + 3 That one has slope 0.5, which is also positive. Here is its graph: Lines that go uphill to the right like this / have positive slopes. Lines that go downhill to the right like this \ have negative slopes. Edwin```
 logarithm/552279: My teacher did not explain any of this stuff to us, so if someone could please EXPLAIN both these types of problums to me that would be awesome. 1) log 7 = 0.8 log 12 = 1.1 log 8= 0.9 Find log 1/64 2)log[5] 11 = 1.5 log[5] 6 = 1.1 log[5] 4 =0.9 Find log[5] 2641 solutions Answer 360213 by AnlytcPhil(1277)   on 2012-01-03 19:01:31 (Show Source): You can put this solution on YOUR website!```logB(A) asks the question: To what exponent must the base B be raised to give A? That exponent is the answer to what logB(A) equals. Because logarithms are exponents, and because we can add exponents of a base in order to multiply, subtract them in order to divide, and multiply them to raise a power to a power, we have these three corresponding rules for logarithms: 1. logB(A·C) = logB(A) + logB(C) 2. logB = logB(A) - logB(C) 3. logB(AC) = C·logB(A) [If the base B isn't written, it's understood to be 10.] -------------------------------------- 1) log(7) = 0.8 log(12) = 1.1 log(8)= 0.9 Find log Use rule 2: 2. logB = logB(A) - logB(C) with A=1, B=64, and the base B understood as 10 log = log(1) - log(64) Now we use the definition of logarithms to find log(1). We ask the question "To what power must the base 10 be raised to get 1?" If you remember that 100 = 1, then you know that log(1) is 0. In fact the logarithm of 1 is always 0, regardless of the base. So now we have log = 0 - log(64) or -log(64) We are given the values of these three logs with understood base 10: log(7) = 0.8, log(12) = 1.1, log(8)= 0.9 Now we ask: Which of these numbers 7, 12, or 8, can we multiply together, divide, or raise to a power, to get 64? The answer: we can get 64 by squaring 8. That is, 8² = 64. So we replace 64 by 8² and now we have: -log(8²) Now we use rule 3: 3. logB(AC) = C·logBA with A=8 and C=2 and we have: -log(82) = -2·log(8) And since we are given log(8) = 0.9 -2·log(8) -2·(0.9) -1.8 That's the answer, -1.8. Try hard to follow the above. -------------------------------------------- 2)log5(11) = 1.5 log5(6) = 1.1 log5(4) = 0.9 Find log5(264) We are given the values of these three logs with base 5: log5(11) = 1.5, log5(6) = 1.1, log5(4) = 0.9 Now we ask: Which of these numbers 11, 6, and 4, can we multiply together, divide, or raise to a power, to get 264? To answer that we must see if 264 can be divided evenly by one of those. We find the 264χ11 = 24. So we write 264 = = 11·24 and we write log5(264) = log5(11·24) Now we use rule 1: 1. logB(A·C) = logB(A) + logB(C) with A=11, B=5, C=24 log5(11·24) = log5(11) + log5(24) We substitute the given log5(11) = 1.5, and we have: 1.5 + log5(24) Now we ask: Which of these numbers 11, 6, and 4, can we multiply together, divide, or raise to a power, to get 24? That answer is easy. 24 = 6·4. So we replace 24 by 6·4, and we have: 1.5 + log5(6·4) Now we use rule 1 again: 1. logB(A·C) = logB(A) + logB(C) this time with with A=6, B=5, C=4 1.5 + log5(6) + log5(4) and we are give those logs, so we substitute: 1.5 + 1.1 + 0.9 Answer: 3.5 --------------------------------------- Edwin```
 Triangles/552267: How can i find prove that both of my triangle are congruent if all of sides are 4 inches for both of my triangles and the each of the angle are 60 degrees 1 solutions Answer 360202 by AnlytcPhil(1277)   on 2012-01-03 17:41:48 (Show Source): You can put this solution on YOUR website!``` Since you're given all corresponding parts to be congruent, you can take your pick of any one of these four theorems: Side-angle-side, Angle-side-angle, Side-angle-side or Side-side-side. ```
 Numbers_Word_Problems/552276: if you multiply this even digit by itself, the sum of the digits of the product is 9. What is the mystery digit?1 solutions Answer 360201 by AnlytcPhil(1277)   on 2012-01-03 17:35:37 (Show Source): You can put this solution on YOUR website!```The digits are 1,2,3,4,5,6,7,8,9,and 0 The even ones are 0,2,4,6,8. The odd ones are 1,3,5,7,9 Let's try the even digit 0. 0Χ0 = 0. There is only one digit, so that can't be it. Let's try the even digit 2. 2Χ2 = 4. There is only one digit, so that can't be it either. Let's try the even digit 8. 8Χ8 = 64. Let's find the sum of the digits 6+4=10. No, 10 is not 9, so it's not 8 I skipped the even digit 6, didn't I? Let's try the even digit 6 6Χ6 = 36 and the sum of its digits is 3+6 or 9. What do you know? That must be the answer, 6. Now was that really too difficult for you to answer all by yourself? Edwin```
 Rational-functions/552200: Solve for x: + = e 1 solutions Answer 360192 by AnlytcPhil(1277)   on 2012-01-03 15:58:29 (Show Source): You can put this solution on YOUR website!``` + = e + = To clear of fractions, multiply through by LCD of + = + = da + bxc = bdxe Get all terms in x on the right by subtracting bxc from both sides: da = bdxe - bxc Factor bx out of the right side: da = bx(de - c) Divide both sides by b(de - c) = = = x x = Edwin```
 Trigonometry-basics/552116: Prove that 1 + tanAtanA/2 = tanAcotA/2 -1 = secA1 solutions Answer 360184 by AnlytcPhil(1277)   on 2012-01-03 15:21:31 (Show Source): You can put this solution on YOUR website!```1 + tan(A)·tan = tan(A)cot - 1 = sec(A) We will make use of the half angle identities: tan = , and its reciprocal cot = , also tan(A) = and sec(A) = First we will prove: 1 + tan(A)·tan = sec(A) 1 + 1 + Get an LCD + Combine fractions over the LCD Cancel first and third terms on top: Cancel the sines: sec(A) --------------------------------- Next we will prove: tan(A)·cot - 1 = sec(A) - 1 - 1 Get an LCD - Combine fractions over the LCD Distribute: Rearrange: Use identity sin²(A)+cos²(A)=1 Cancel 1-cos(A)'s sec(A) Edwin```
 Mixture_Word_Problems/552163: Please help me solve this mixture problem: How many pounds of gourmet candy selling for \$2.80 per pound should be mixed with 7 pounds of gourmet candy selling for \$1.60 per pound to obtain a mixture selling for \$1.96 per pound? A. 4 lb B. 3 lb C. 1 lb D. 5 lb I can solve it by trial and error, but is their a formula that will work?1 solutions Answer 360161 by AnlytcPhil(1277)   on 2012-01-03 14:01:24 (Show Source): You can put this solution on YOUR website!How many pounds of gourmet candy selling for \$2.80 per pound should be mixed with 7 pounds of gourmet candy selling for \$1.60 per pound to obtain a mixture selling for \$1.96 per pound? ``` + = ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ \$2.80x + \$11.20 = \$1.96(x+7) Drop the dollar marks and move the decimals two places right 280x + 1120 = 196(x+7) 280x + 1120 = 196x + 1372 Subtract 1120 from both sides: 280x = 196x + 252 Subtract 196x from both sides: 84x = 252 Divide both sides by 84 = x = 3 Answer 3 pounds. Edwin```
 Trigonometry-basics/552130: Prove the following identity: (1 + sin x + cos x)/(1 + sin x - cos x) = cot(x/2)1 solutions Answer 360147 by AnlytcPhil(1277)   on 2012-01-03 12:38:44 (Show Source): You can put this solution on YOUR website!```A familiar identity is tan = , and since the cotangent is the reciprocal of the tangent, we have cot = , so we will try to make the left side into that expression. So we will multiply the left side by , so hopefully it will have the desired denominator in the end. 1 + sin(x) + cos(x)  = cot() 1 + sin(x) - cos(x) [1 + sin(x) + cos(x)] [1 - cos(x)]  ·  [1 + sin(x) - cos(x)] [1 - cos(x)] 1 - cos(x) + sin(x) - sin(x)cos(x) + cos(x) - cos²(x)  [1 + sin(x) - cos(x)][1 - cos(x)] 1 - cos(x) + sin(x) - sin(x)cos(x) + cos(x) - cos²(x)  [1 + sin(x) - cos(x)][1 - cos(x)] 1 + sin(x) - sin(x)cos(x) - cos²(x)  [1 + sin(x) - cos(x)][1 - cos(x)] Rearrange the terms 1 - cos²(x) + sin(x) - sin(x)cos(x)  [1 + sin(x) - cos(x)][1 - cos(x)] [1 - cos²(x)] + sin(x) - sin(x)cos(x)  [1 + sin(x) - cos(x)][1 - cos(x)] Use the identity sin²(x) = 1 - cos²(x) sin²(x) + sin(x) - sin(x)cos(x)  [1 + sin(x) - cos(x)][1 - cos(x)] Factor out sin(x) on the top: sin(x)[sin(x) + 1 - cos(x)]  [1 + sin(x) - cos(x)][1 - cos(x)] Then we can cancel: sin(x)[sin(x) + 1 - cos(x)]  [1 + sin(x) - cos(x)][1 - cos(x)] sin(x)  1 - cos(x) which is the identity we showed above for cot() cot Edwin```
 Trigonometry-basics/552128: Prove the following identity: 2sin(x+y)sin(x-y) = cos2y-cos2x1 solutions Answer 360139 by AnlytcPhil(1277)   on 2012-01-03 10:54:58 (Show Source): You can put this solution on YOUR website!```This is one tough identity! It requires very unusual substitutions, additions and subtractions of the same quantities. 2sin(x+y)sin(x-y) = cos(2y)-cos(2x) = cos(y+y) - cos(x+x) = cos(y+y+x-x) - cos(x+x+y-y) = cos[(x+y)-(x-y)] - cos[(x+y)+(x-y)] Use the identities cos(A∓B)=cos(A)cos(B)±sin(A)sin(B) with A=(x+y), B=(x-y) = [cos(x+y)cos(x-y)+sin(x+y)sin(x-y)] - [cos(x+y)cos(x-y)-sin(x+y)sin(x-y)] = cos(x+y)cos(x-y) + sin(x+y)sin(x-y) - cos(x+y)cos(x-y) + sin(x+y)sin(x-y) = cos(x+y)cos(x-y) + sin(x+y)sin(x-y) - cos(x+y)cos(x-y) + sin(x+y)sin(x-y) = 2sin(x+y)(sin(x-y) Edwin```
 logarithm/552114: Solve the equations: 4log4x - 3log5y =1 log2x - log5y =21 solutions Answer 360133 by AnlytcPhil(1277)   on 2012-01-03 09:43:13 (Show Source): You can put this solution on YOUR website!```I can't tell which of these you meant: 4log(4x) - 3log(5y) = 1 or 4log4(x) - 3log5(y) = 1 log(2x) - log(5y) = 2 log2(x) - log5(y) = 2 I tried doing it the first way and there was no solution, so I think you must have meant the second way. 4log4(x) - 3log5(y) = 1 log2(x) - log5(y) = 2 I will change the first term in the first equation so it will be a log to the base 2, like the first term in the second equation. Using the change of base formula on that term: 4log4(x) = = = = = 2log2(x) The system of equations is now: 2log2(x) - 3log5(y) = 1 log2(x) - log5(y) = 2 let u = log2(x) let v = log5(y) The system of equations is now: 2u - 3v = 1 u - v = 2 Solve that system of equations by substitution or elimination and get (u,v) = (5,3). I'm sure you can do that. But we don't want u and v, we want x and y. So we substitute back: u = log2(x), which is equivalent to the exponential equation: x = 2u = 25 = 32 v = log5(y) y = 5v = 53 = 125 So the solution is (x,y) = (32,125) Edwin```
 Polynomials-and-rational-expressions/552015: Factor out the greatest common factor from the expression 10x^2yz^3  6x^2y^2z  4xy^3z^2 1 solutions Answer 360063 by AnlytcPhil(1277)   on 2012-01-02 23:01:20 (Show Source): You can put this solution on YOUR website!```10x²y³z³  6x²y²z  4xy³z² 10, 6, and 4 can all be divided by 2, so we can take out a 2 factor x², x², and x can all be divided evenly by x, so we can take out an x factor y³, y², and y³ can all be divided evenly by y², so we can take out a y² factor z³, z, and z² can all be divided evenly by z, so we can take out a z factor So we can take out 2xy²z. So we write this: 2xy²z( To get the first term we divide 10x²y³z³ by 2xy²z and get 5xyz², so we write that first in the parentheses: 2xy²z(5xyz² To get the second term we divide 6x²y²z by 2xy²z and get -3x, so we write that second in the parentheses: 2xy²z(5xyz² - 3x To get the second term we divide 4xy³z² by 2xy²z and get -2yz, so we write that third in the parentheses and close the parentheses: 2xy²z(5xyz² - 3x - 2yz) Edwin```
 Triangles/552045: in the accompanying diagram of Triangle ABC, AB is congruent to AC. The measure of Angle B is 40 degrees. What is the measure of Angle A?1 solutions Answer 360052 by AnlytcPhil(1277)   on 2012-01-02 22:36:12 (Show Source): You can put this solution on YOUR website!``` Since AB is congruent to AC, we know that ᐃABC is isosceles. Therefore we know that its base angles are congruent and thus have the same measure. Therefore we will label the measure of the other base angle C, m∠A = 40° as well: Since we know that the sum of the measures of the three angles of any triangle must always equal to 180°, we can write m∠A + m∠B + m∠C = 180° m∠A + 40° + 40° = 180° m∠A + 80° = 180° m∠A = 180° - 80° m∠A = 100° Edwin```
 Divisibility_and_Prime_Numbers/551088: what are the steps and answer to 450 divided by 2701 solutions Answer 359360 by AnlytcPhil(1277)   on 2011-12-30 12:30:04 (Show Source): You can put this solution on YOUR website!```450 divided by 270 Here are two ways: First way: 1 270)450 270 180 = = Second way: Since both end with a 0, they can both be divided by 10, so: To divide by 10, just drop off the 0's on the end and you get Now if we remember our 9's multiflication facts, we remember that 5Χ9 = 45 and 3Χ9 = 27, so and you get Then you change that to a mixed number and get: Edwin```
 Circles/551080: how i do this? find the equation of a circle centered at the origin, and having radius 1 find the equation of a circle centered at the origin and having radius 5 find the equation of a circle centered at the origin, and having radius ,/19 find the equation of a circle centered at the origin and having radius 3,/7 1 solutions Answer 359359 by AnlytcPhil(1277)   on 2011-12-30 12:17:58 (Show Source): You can put this solution on YOUR website!```Substitute the value of the radius for r in x² + y² = r² For instance, to get the answer to ``` find the equation of a circle centered at the origin and having radius 5 ```just substitute 5 for r and get x² + y² = 5² and then do one more step, change 5² to 25 x² + y² = 25 That's the equation. That's all there is to it! Edwin```
 Rate-of-work-word-problems/551064: Three friends a,b,c can do a piece of work in T hours working together. working alone, a can do the work in 6 hours more, b in 1 hour more and c in twice the time if all of them were working together. How long would it take to finish the work if all of them were working together (find the value of T)?1 solutions Answer 359355 by AnlytcPhil(1277)   on 2011-12-30 11:59:00 (Show Source): You can put this solution on YOUR website!Three friends a,b,c can do a piece of work in T hours working together. working alone, a can do the work in 6 hours more, b in 1 hour more and c in twice the time if all of them were working together. How long would it take to finish the work if all of them were working together (find the value of T)? ```Make this chart. number of number of time in pieces of hours pieces of work required work/hour a working alone b working alone c working alone all 3 working together In each case exactly 1 piece of work is completed, so we put 1 for the number of pieces of work in all 4 cases. We also put in T for the time for all three working together, T+6 hours for a's time, T+1 hours for b's time, and 2T for c's time. number of number of rate in pieces of hours pieces of work required work/hour a working alone 1 T+6 b working alone 1 T+1 c working alone 1 2T all 3 working together 1 T We fill in the rates in pieces of work/hour by dividing the number of pieces of work by the hours required: number of number of rate in pieces of hours pieces of work required work/hour a working alone 1 T+6 b working alone 1 T+1 c working alone 1 2T all 3 working together 1 T The equation comes from: + + = + + = Solve that equation and get T = of an hour or 40 minutes. Edwin```
 Triangles/551038: Find the centre and circumradius of a triangle with vertices A(4,3),B(-3,2) and C(1,-6).1 solutions Answer 359349 by AnlytcPhil(1277)   on 2011-12-30 11:30:40 (Show Source): You can put this solution on YOUR website!``` Let the coordinates of the circumcenter (or "circumcentre", as you spell it in the UK) be O(h,k). Then the circumradius r = AO = BO = CO. We use the distance formula d = AO = BO = = CO = = = = Squaring all three expressions: (h-4)² + (k-3)² = (h+3)² + (k-2)² = (h-1)² + (k+6)² Equating the first two: (h-4)² + (k-3)² = (h+3)² + (k-2)² Rearrange terms to create the difference of squares: (h-4)² - (h+3)² = (k-2)² - (k-3)² Factor (or as they say in UK, "factorise") [(h-4) - (h+3)][(h-4) + (h+3)] = [(k-2) - (k-3)][(k-2) + (k-3)] [h - 4 - h - 3][h - 4 + h + 3] = [k - 2 - k + 3][k - 2 + k - 3] [-7][2h-1] = [1][2k-5] -14h + 7 = 2k - 5 -14h = 2k - 12 -7h = k - 6 Equating the lst and 3rd (h-4)² + (k-3)² = (h-1)² + (k+6)² Rearrange terms to create the difference of squares: (h-4)² - (h-1)² = (k+6)² - (k-3)² Factor ("factorise") [(h-4) - (h-1)][(h-4) + (h-1)] = [(k+6) - (k-3)][(k+6) + (k-3)] [h - 4 - h + 1][h - 4 + h - 1] = [k + 6 - k + 3][k + 6 + k - 3] [-3][2h-5] = [9][2k+3] -6h + 15 = 18k + 27 -6h = 18k + 12 h = -3k - 2 So we solve this system of two equations: -7h = k - 6 h = -3k - 2 by substitution and get -7(-3k - 2) = k - 6 21k + 14 = k - 6 20k = -20 k = -1 h = -3(-1) - 2 h = 3 - 2 h = 1 So the circumcenter (or "circumcentre") is O(h,k) = O(1,-1) The circumradius r is r = AO = = = = = = 5 Edwin```
 Mixture_Word_Problems/551049: A goat is tied to one of the corners of a rectangular barn on a rope that is 50 feet long. The dimensions of the barn are 40 feet by 30 feet. Assuming that the goat can graze wherever its rope allows it to reach, what is the square footage of the grazing area for the goat?1 solutions Answer 359324 by AnlytcPhil(1277)   on 2011-12-30 08:21:00 (Show Source): You can put this solution on YOUR website!```A goat is tied to one of the corners of a rectangular barn on a rope that is 50 feet long. The dimensions of the barn are 40 feet by 30 feet. Assuming that the goat can graze wherever its rope allows it to reach, what is the square footage of the grazing area for the goat? The area of a circle is pr² The grazing area consists of 1. three quarters of a big 50ft-radius circle, which has area ·p(50)² = ·2500p square feet = 1875p square feet. 2. one quarter of a 20ft-radius circle on the left, which has area ·p(20)² = ·400p square feet = 100p square feet 3. one quarter of a small 10ft-radius circle on the top, which has area ·p(10)² = ·100p square feet = 25p square feet. Total = 1875p + 100p + 25p = 2000p square feet of grazing area. Edwin McCravy aka AnlytcPhil```
 Triangles/546959: Can the base of an isosceles triangle be shorter than the leg of the triangle? Please explain why. Need to know for Finals! Thanks 1 solutions Answer 356156 by AnlytcPhil(1277)   on 2011-12-13 21:17:01 (Show Source): You can put this solution on YOUR website!The triangle below is isosceles. Look how short the base is and look how long the legs are!
 Word_Problems_With_Coins/546917: I have twenty four coins out of the twenty four coins how many are nickels with the total of \$4.80 1 solutions Answer 356148 by AnlytcPhil(1277)   on 2011-12-13 21:05:13 (Show Source): You can put this solution on YOUR website!```There are many answers. You could have 3 nickels like this: 17 quarters = \$4.25 4 dimes .40 3 nickels = .15 ------------------- 24 coins = \$4.80 Or you could have 6 nickels like this: 18 quarters = \$4.50 6 nickels = .30 ------------------- 24 coins = \$4.80 Or you could have 0 nickels at all 19 quarters = \$4.75 5 pennies = .05 ------------------- 24 coins = \$4.80 If half dollars or dollar coins are allowed there are many many more ways. Here is a way with 4 nickels if you use 8 half dollars. 8 half dollars = \$4.00 2 quarters = .50 4 nickels = .20 10 pennies = .10 ----------------------- 24 coins = \$4.80 If you can use dollar coins, here's a way to have 17 nickels. 3 dollar coins = \$3.00 1 half dollar = .50 1 quarters = .25 2 dimes = .20 17 nickels = .85 ----------------------- 24 coins = \$4.80 Edwin```
 Miscellaneous_Word_Problems/546910: The number "N" is a 5 digit natural number. The 6 digit number N1, formed by placing the digit 1 after N, is 3 times as large as the 6 digit number 1N, formed by placing the 1 in front of N. What is the original 5 digit number N? Write each as an algebraic expression: N1 = ? and 1N = ? Write the equation to solve for finding "N"=1 solutions Answer 356122 by AnlytcPhil(1277)   on 2011-12-13 19:58:54 (Show Source): You can put this solution on YOUR website!The number "N" is a 5 digit natural number. The 6 digit number N1, formed by placing the digit 1 after N, is 3 times as large as the 6 digit number 1N, formed by placing the 1 in front of N. What is the original 5 digit number N? Write each as an algebraic expression: N1 = ? and 1N = ? Write the equation to solve for finding "N"= ``` N = ↏↏↏↏↏ N1 = ↏↏↏↏↏1 = 10N + 1 1N = 1↏↏↏↏↏ = 100000 + N 10N + 1 = 3(100000 + N) 10N + 1 = 300000 + 3N 7N = 299999 N = 42857 N1 = 428571 1N = 142857 Edwin```
 Radicals/541279: Please help me solve and explain so I can help my son understand. 3 - 2 radical 11 / 2 + radical 11 Write the equation in simplified radical form.1 solutions Answer 354134 by AnlytcPhil(1277)   on 2011-12-04 12:28:28 (Show Source): You can put this solution on YOUR website!``` Form the conjugate of the denominator by the rule: the conjugate of A + B is A - B. You just change the sign of the second term but keep the sign of the first term. So the conjugate of the denominator is . Put this conjugate over itself like this: . Since that red fraction just equals 1 we can multiply our express by it and not change its numerical value, but change its form. So we multiply by it: Χ. We'll put parentheses around the binomials: Χ. And multiply numerators and denominators: To get the negative sign off the bottom, bring it to the top: We can multiply the negative sign into the parentheses so it won't be sticking out in front: Finally we will reverse the terms in the parentheses: That's the simplest form. Or this way Edwin```
 Polynomials-and-rational-expressions/541273: State the degree of the following polynomial: -6x²y + 4xy  7y how would i go about solving this 1 solutions Answer 354131 by AnlytcPhil(1277)   on 2011-12-04 12:05:25 (Show Source): You can put this solution on YOUR website!``` -6x²y + 4xy  7y Get rid of the exponents by writing the square x² as the product of x and x. -6xxy + 4xy - 7y Now count the number of letters multiplied by the coefficient in each of the three terms: -6xxy has a row of 3 letters multiplied by the coefficient -6, so that term has degree 3. +4xy has a row of 2 letters multiplied by the coefficient +4, so that term has degree 2. -7y has just one letter multiplied by the coefficient -7, so that term has degree 1. Now whichever term has the largest degree, we take that to be the degree of the ENTIRE polynomial. So since the first term has the largest degree, which is 3, the whole polynomial has that same degree, which is 3. In a nutshell, the degree of a polynomial is the most number of letters multiplied together by the coefficient in any of its terms. An exponent is considered to be a string of the same letter repeated to be multiplied the same number of times as the exponent. The longest string of letters in any of the three terms is 3, so the polynomial, which is called a "trinomial" because it has three terms, has degree 3. But don't get those mixed up just because they're both 3. It's a trinomial because it has three terms, but it has degree 3 because the first term has the most number of letters multiplied together by the coefficient, which is 3 letters. Edwin```
 Polygons/541201: a regular polygon has 39 sides, what is the size of each angle?1 solutions Answer 354102 by AnlytcPhil(1277)   on 2011-12-04 06:17:44 (Show Source): You can put this solution on YOUR website! = = = °
 Permutations/540987: A credit card number has 5 digits (between 1 to 9). The first two digits are 12 in that order, the third digit is bigger than 6, and the fourth digit is 3 times the 5th digit. How many diff erent credit card combinations are possible?1 solutions Answer 354027 by AnlytcPhil(1277)   on 2011-12-03 19:25:59 (Show Source): You can put this solution on YOUR website!```Since the first two digits are 12 in that order, there is only 1 way to choose the first two digits. That's 1 way to choose the first two digits. Since the third digit is bigger than 6, it can only be 7,8, or 9. So that's 1Χ3 choices for the the first three digits. Since the 4th digit is 3 times the 5th digit, the number can only end in 31, 62, or 93 So for each of the 1Χ3 way to choose the first 3 digits, there are 3 ways to choose the last two digits. That's a total of 1Χ3Χ3 = 9 possible numbers. Here are all 9: 1. 12731 2. 12762 3. 12793 4. 12831 5. 12862 6. 12893 7. 12931 8. 12962 9. 12993 Edwin```
 Mixture_Word_Problems/541055: Hello can someone please assist me with the correct approach to solving this problem? I would greatly appreciate it. Thanks in advance. Candy Mixtures - Someone wants to mix some candy that is worth 45 cents per pound. Some is worth 80 cents per pound to make 350 lb of a mixture worth 65 cents per pound. How much of each type of candy should be used? 1 solutions Answer 354023 by AnlytcPhil(1277)   on 2011-12-03 19:02:43 (Show Source): You can put this solution on YOUR website!Candy Mixtures - Someone wants to mix some candy that is worth 45 cents per pound. Some is worth 80 cents per pound to make 350 lb of a mixture worth 65 cents per pound. How much of each type of candy should be used? ```Suppose we mix x pounds of the 45’ candy with y pounds of the 80’ candy. Then we have two equations, a candy equation and a money equation: The candy equation comes from this: + = x + y = 350 The money equation comes from + = (45’)x + (80’)y = (60’)(350) or 45x + 80y = 21000 So solve this system of equations: x + y = 350 45x + 80y = 21000 (If you can't solve that system of equations, post again asking how) Answer: Mix 200 pounds of the less expensive candy and 150 pounds of the cheaper candy. Edwin```