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# Recent problems solved by 'richard1234'

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 test/549517: 2x +6y1 solutions Answer 358526 by richard1234(5390)   on 2011-12-25 23:07:29 (Show Source): You can put this solution on YOUR website!Do you have a question?
 Polynomials-and-rational-expressions/550122: Essay: Show all work. The area of a rectangular swimming pool is given by 2x^2 - 3x -9 ft^2. One side length of the pool is given by 2x + 3 feet. What is an algebraic expression for the other side length of the pool? Simplify this, and include correct units as part of your answer. 1 solutions Answer 358525 by richard1234(5390)   on 2011-12-25 23:06:40 (Show Source): You can put this solution on YOUR website!The other side length is equal to I'll let you do the polynomial division. You should get x - 3 as your result. The units are in feet (since we are looking for a length).
 Reduction-of-unit-multipliers/550197: Is Celsius temperature a function of Fahrenheit temperature? explain. please answer this...thanks1 solutions Answer 358524 by richard1234(5390)   on 2011-12-25 23:05:12 (Show Source): You can put this solution on YOUR website!Yes, we have C = (5/9)(F - 32) where F is the temperature in degrees Fahrenheit.
 Functions/550132: How many positive integers n are such that the value of the expression n^3 -14n^2 + 64n - 93 is a prime number?1 solutions Answer 358522 by richard1234(5390)   on 2011-12-25 23:00:25 (Show Source): You can put this solution on YOUR website!The expression factors to (n-3)(n^2 - 11n + 31). Since we want a prime number, either n-3 or n^2 - 11n + 31 must be equal to 1 (or -1). Solving for n, we obtain n = 2, 4, 5, or 6. Here are the values we obtain when we replace n into the given expression: n = 2 --> -13 n = 4 --> 3 n = 5 --> 2 n = 6 --> 3 The values n = 4, 5, 6 produce prime numbers, so the answer is 3.
 Equations/550195: i do not understand how to do one step equatons? can you explain to me how to do them?1 solutions Answer 358521 by richard1234(5390)   on 2011-12-25 22:55:46 (Show Source): You can put this solution on YOUR website!If you're trying to solve for x, do whatever operation is necessary to isolate x. Note the phrase "one-step equation," should only take one step to solve, as opposed to computer algorithms which may take hundreds of steps.
 Sequences-and-series/550089: prove that if k>1 then k^n→∞ an n→∞ there is a hint given. Hint:let k=1+t where t>0 and use the fact that (1+t)^n>1+nt prove that if k is between 0 and 1 then k^n tends to 0 as n tends to infinity1 solutions Answer 358405 by richard1234(5390)   on 2011-12-24 12:07:46 (Show Source): You can put this solution on YOUR website!Given the hint, the problem becomes simple. Since 1 + nt approaches infinity as n approaches infinity, the LHS of the inequality is "bounded" by this expression, so the LHS must also approach infinity.
 Permutations/549700: Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for the results, each member of the group should be prepared to respond. Minimum Wage The table shows the minimum wage for three different years. (a) Make a scatterplot of the data in the viewing rectangle [1930, 2010, 10] by [0, 6, 1]. (b) Find a quadratic function given by that models the data. (c) Estimate the minimum wage in 1976 and compare it to the actual value of $2.30. (d) Estimate when the minimum wage was$1.00. (e) If current trends continue, predict the minimum wage in 2009. Compare it to the projected value of \$7.25.1 solutions Answer 358371 by richard1234(5390)   on 2011-12-24 02:04:36 (Show Source): You can put this solution on YOUR website!No data provided. Also, do you expect tutors on algebra.com to form groups of 2 to 4 people to answer these questions?
 Triangles/549465: As the legs of a tripod are spread apart, which theorem guarantees that the angles between each pair of legs get larger?1 solutions Answer 358369 by richard1234(5390)   on 2011-12-24 02:02:35 (Show Source): You can put this solution on YOUR website!The law of cosines is sufficient. If d is the distance between two ends of the tripod (which is assumed to be increasing), and x is the length of each leg, then where theta is the angle between two of the legs. It can be shown that if d increases, then theta also increases.
 Linear-equations/549947: Explain why a point on the boundary line can not be used as a test point when grapging a linear inequality in two variables? provide an example.1 solutions Answer 358366 by richard1234(5390)   on 2011-12-24 01:44:50 (Show Source): You can put this solution on YOUR website!Points on the boundary line represent the equality cases, or where the two expressions you are comparing are equal. For example, if you know that x + y > 5 (note "strictly greater than 5") and you plug in (3,2), you get 5 > 5, which is not true. However if the inequality were x + y >= 5, (3,2) works.
 Sequences-and-series/549996: Can a function be continuous at one value of x and discontinuous at all other x E R? Explain your answer, giving proofs where appropriate. 1 solutions Answer 358365 by richard1234(5390)   on 2011-12-24 01:39:38 (Show Source): You can put this solution on YOUR website!No. If f is continuous at x and only x, then for any x-values arbitrarily close to x, f must also be continuous (since continuity is defined using the limit as x approaches a fixed number), so we have a contradiction.
 Miscellaneous_Word_Problems/548989: What is the least positive integral value of n for which (n - 12) divided by (5n + 23) is a non-zero reducible fraction? 1 solutions Answer 358234 by richard1234(5390)   on 2011-12-23 02:20:48 (Show Source): You can put this solution on YOUR website!Assume n > 12 so that the fraction is positive (1 through 11 don't work anyway). If the fraction is reducible for some n, then n-12 and 5n+23 must have a common factor p (other than 1). We can write this using modular arithmetic: (mod p) This implies (mod p), so (mod p). Hence, p = 83. Therefore, we can set n = 95, obtaining the fraction 83/498, or 1/6.
 Sequences-and-series/549787: Give an example of a function that is continuous at all non-integer values but is discontinuous at all integer values. Prove the discontinuity property of your function. 1 solutions Answer 358233 by richard1234(5390)   on 2011-12-23 02:07:53 (Show Source): You can put this solution on YOUR website!For example, let f(x) = 1 if x is not an integer, but f(x) = 0 if x is an integer (it is easier to write this function using piece-wise notation). A function f(x) is not continuous if (but not only if) . This is true because if a is an integer, the LHS equals 1 but the RHS equals 0. Therefore f is not continuous.
 Volume/549854: determine the volume of the three-dimensional figure. When appropriate, round your answer to the nearest hundredth.a)12 in., b)30 in. ,c)30 in.2 1 solutions Answer 358232 by richard1234(5390)   on 2011-12-23 02:02:44 (Show Source): You can put this solution on YOUR website!You didn't provide any information whatsoever of the figure. Also, none of the answer choices can be correct because the units for volume are (length)^3.
 decimal-numbers/549955: Sum of three positive integers is 2^2011 + 1, and the product of two of them is 2^2011.how many values can the third integer take?1 solutions Answer 358230 by richard1234(5390)   on 2011-12-23 01:52:27 (Show Source): You can put this solution on YOUR website!Since the product of two of these integers is a power of 2, it follows that both integers are powers of 2. Hence these integers can be {1, 2^2011}, {2, 2^2010}, ..., {2^1005, 2^1006} (order doesn't matter). The two integers cannot be {1, 2^2011} since the third integer would be zero. However, each of the other ordered pairs produces a unique integer (e.g. subtract the sum of the two integers from 2^2011 + 1), so the number of values for the third integer is the number of ordered pairs in {2, 2^2010}, ..., {2^1005, 2^1006}, or 1004 integers.
 Triangles/548749: The centroid of a triangle is ____?____ the circumcenter of a triangle. A. Always B. Sometimes C. Never1 solutions Answer 358228 by richard1234(5390)   on 2011-12-23 01:10:21 (Show Source): You can put this solution on YOUR website!B, sometimes. The centroid is defined as the "center of mass" of the triangle, and the circumcenter is not defined this way. However, they can still be the same point, as in the case of an equilateral triangle.
 Parallelograms/549542: Which of the following conditions or set of conditions is sufficient for a parallelogram to be a square? 1 solutions Answer 358227 by richard1234(5390)   on 2011-12-23 01:07:39 (Show Source): You can put this solution on YOUR website!The only conditions required are that the parallelogram has four congruent sides (or equivalently, equal diagonals) and four right angles.
 Polynomials-and-rational-expressions/548823: Essay. List the terms, coefficients, and degree of the following polynomial: 5x^8 + 3x^6 – 2x + 4/5 or four fifths.1 solutions Answer 358225 by richard1234(5390)   on 2011-12-23 01:00:54 (Show Source): You can put this solution on YOUR website!Terms: 5x^8, 3x^6, -2x, 4/5 Coefficients: 5, 3, -2, 4/5 Degree: 8 (highest exponent of x) I don't know why the problem requires an essay...
 sets-and-operations/549952: given the universal set E={1,2,3,4,5,6,7,8,9,10} what is the complement of {1,10} 1 solutions Answer 358224 by richard1234(5390)   on 2011-12-23 00:57:27 (Show Source): You can put this solution on YOUR website!{2,3,4,5,6,7,8,9}
 Quadratic_Equations/549912: select a natural number between 1 to 50. square the digits and add. repeat this process until you see a pattern. What other numbers end with one? What happen if you do the process to number 37? What conclusion can you give?1 solutions Answer 358223 by richard1234(5390)   on 2011-12-23 00:56:53 (Show Source): You can put this solution on YOUR website!Suppose I choose 4. The numbers in the pattern are 16, 37, 58, 89, 145, 42, 20, 4, 16, and it repeats. If the number 1 appears in the sequence, then the previous number must have been 1, 10, 100 or any power of 10. However, all succeeding terms in the sequence must be 1. Doing the process to the number 37 yields the same sequence as if I did 4 (up to shifting terms). We can conjecture that every number will eventually produce a periodic sequence. However we want to prove this. Given a sequence a_1, a_2, ... we want to show that a_i = a_j for some distinct i,j (this will produce a periodic sequence). Since it is tricky to establish an explicit formula for a_i, we can instead show that a_i attains a maximum value. Since your sequence can be arbitrarily long, then by Pigeonhole principle, two numbers in the sequence must be the same, and hence the sequence is periodic. I'll leave it to you to prove that a_i attains a maximum value...
 Sequences-and-series/549782: Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite “sum to infinity” exists if and only if r < 1 and show that in this case the sum to infinity is . 1 solutions Answer 358222 by richard1234(5390)   on 2011-12-23 00:44:56 (Show Source): You can put this solution on YOUR website!We prove that a convergent sum exists by evaluating the limit as the number of terms approaches infinity: Since r^(n+1) tends to zero, the sum converges to 1/(1-r). However, this only holds when |r| < 1. Otherwise, the limit diverges.
 Equations/549950: 2x + 3y =1 -2x + y =111 solutions Answer 358220 by richard1234(5390)   on 2011-12-23 00:38:26 (Show Source): You can put this solution on YOUR website!Add both equations to get 4y = 12; this implies y = 3. Substitute y with 3 into either equation to obtain x = -4.
 Quadratic_Equations/548600: The solutions of x^3 – 2x^2 + 3x + 4 = 0 are a, b, and c. what is the numerical value of (a + b)(a + c)(b + c) ? 1 solutions Answer 358053 by richard1234(5390)   on 2011-12-22 11:37:11 (Show Source): You can put this solution on YOUR website!By Vieta's formulas, a+b+c = 2. Hence we are evaluating (2-c)(2-b)(2-a). We could expand this entire expression (since we are guaranteed to obtain useful cyclic sums: However, we have a+b+c = 2, ab+ac+bc = 3, abc = 4 (you can verify these by expanding (x-a)(x-b)(x-c) and equating like terms). Hence, the expression we want is equal to
 Triangles/548650: in an isocles triangle, without any measurements, i have to find the each individual angle, what is a formula i could use?1 solutions Answer 358048 by richard1234(5390)   on 2011-12-22 11:31:11 (Show Source): You can put this solution on YOUR website!You need to know at least one of the angles (or some side lengths), and use the constraint that the sum of the angles is 180.
 Polynomials-and-rational-expressions/549780: Short Answer. The length of the side of a rectangular desk is given by x - 4y cm. The length of the front is given by x + 4y cm. What is an algebraic expression for the area of the top of the rectangular desk? Include correct units as part of your answer.1 solutions Answer 358045 by richard1234(5390)   on 2011-12-22 11:28:58 (Show Source): You can put this solution on YOUR website!The area is (x-4y)(x+4y) = x^2 - 16y^2 (cm^2).
 Sequences-and-series/549775: Let k > 0 be a constant and consider the important sequence {kn}. It’s behaviour as n tends to infinity will depend on the value of k. (i) State the behaviour of the sequence as n tends to infinity when k = 1 and when k = 0. (ii) Prove that if k > 1 then kn tends to infinity as n teds to infinity (hint: let k = 1 + t where t > 0 and use the fact that (1 + t)n > 1 + nt. (iii) Prove that if 0 < k < 1 then kn tends to 0 as n tends to infinity . 1 solutions Answer 358044 by richard1234(5390)   on 2011-12-22 11:18:18 (Show Source): You can put this solution on YOUR website!Is your sequence kn or k^n? You typed kn, which is interpreted as "k times n" but in question iii) but you implied that the limit as n goes to infinity of kn/k^n (where 0 < k < 1) is zero. Either way, you can use limits or any other technique. If you mean k^n, part i) is simple, because 1^n is always 1 and 0^n is always 0. For part ii), we can actually prove it by assuming that on the other hand k^n converges to some number X. If this is so, then k(k^n) must also converge (to kX). However, k(k^n) is equivalent to k^n (since we are evaluating where n approaches infinity) so kX = X. This implies X = 0 (since k is not 1), contradiction. Hence k^n diverges. For part iii), assume on the other hand it diverges. We have |k(k^n)| < |k^n| (since |k| < 1). This is also a contradiction because we assumed it diverges, so it converges. By the same argument in ii), it converges to 0.
 Divisibility_and_Prime_Numbers/549506: What is the remainder when 2^1024 + 5^1024 +1 is divided by 9?1 solutions Answer 358040 by richard1234(5390)   on 2011-12-22 11:03:37 (Show Source): You can put this solution on YOUR website!Several ways to do this. One way is to that , and that (modulo 9). Similarly, (modulo 9). Hence the sum is equivalent to 7 + 4 + 1 = 12 = 3 (modulo 9), so the remainder is 3.
 Geometry_proofs/548583: please give me a short definiton of prrof that referrs to geometry Please1 solutions Answer 357117 by richard1234(5390)   on 2011-12-18 10:13:36 (Show Source): You can put this solution on YOUR website!A proof is simply a set of statements that proves why a certain theorem or statement is true.
 Rectangles/548607: what is the perimeter of a triangle whose length is 7 feet and whose width is 6 feet1 solutions Answer 357115 by richard1234(5390)   on 2011-12-18 10:10:09 (Show Source): You can put this solution on YOUR website!A triangle? Or rectangle?
 Polynomials-and-rational-expressions/547904: Take a number (except 1). Square that number and then subtract 1. Divide by one less than your original number. Now subtract your original number. How does this number game work? How did the number game use the skill of simplifying rational expressions? 1 solutions Answer 357057 by richard1234(5390)   on 2011-12-17 22:23:23 (Show Source): You can put this solution on YOUR website!You'll always get 1 because
 Exponents-negative-and-fractional/548528: Essay; show all work. Consider that the age, x, of a unicorn in human equivalent years can be given by the formula f(x) = - 0.011718x4 + 0.187326x3 – 1.3367x2 + 12.46x + 2.914. When a unicorn is 2.5 years old, what is its age in human equivalent years? What about when it is 12 years old? 1 solutions Answer 357056 by richard1234(5390)   on 2011-12-17 22:20:11 (Show Source): You can put this solution on YOUR website!Replace x with 2.5, then simplify to find f(2.5). Do the same with f(12).