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# Recent problems solved by 'jsmallt9'

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 logarithm/706157: how to arrive at the log of -6.831 solutions Answer 435006 by jsmallt9(3296)   on 2013-01-25 08:10:29 (Show Source): You can put this solution on YOUR website!log(-6.83) does not exist. It is impossible to raise 10, the base of "log", to any power and get a negative result like -6.83.
 Numbers_Word_Problems/706129: How many 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit? 1 solutions Answer 435004 by jsmallt9(3296)   on 2013-01-25 08:06:00 (Show Source): You can put this solution on YOUR website!Let's start by listing these numbers in an organized way. We may be able to see a pattern that will help us find the answer quickly. (If we don't find a pattern them we can just list them all and count.) Numbers do not start with a zero (that counts) so we will start with the two-digit numbers that start with 1: ```How many Two-digit numbers 1 10 ``` All the other two-digit numbers that start with 1 will have a units/ones digit that is not less than the tens digit. Now we'll add the numbers that start with 2's and 3's: ```How many Two-digit numbers 1 10 2 20 21 3 30 31 32 ``` At this point we may see a pattern that helps. We can see that the first row has 1 number, the 2nd row has two numbers, the 3rd row has 3 numbers. So our last row, the 9th, will have nine numbers. You may have learned a formula for the sum of consecutive Integers: n*(n+1)/2. Since we will have 9 numbers to add this would be: 9*(9+1)/2 = 9*10/2 = 45. If you don't know about this formula then picture this:The numbers are forming a right triangle as we go down the list.Imagine another triangle just like the one our list makes.Imagine this 2nd triangle flipped upside down so that the longest row (the numbers in the nineties) is on the top.Now imagine the two triangles next to each other so that the "10" of the first triangle is in front of the 9 nineties numbers from the flipped triangle; the "20 21" is in front of the 8 eighties numbers fron the flipped triangle; and all the way down the the 9 nineties numbers from the first triangle in front of the "10" of the flipped triangle.The combined shape of the two triangles should be a rectangle with 9 rows and 10 numbers in each row. This means there are 9*10 or ninety numbers in this rectangle.But this rectangle has two copies of every number, one from each triangle. So the count of the numbers in just one triangle is going to be 1/2 of the total for the rectangle: 90/2 = 45. Note how we just figured out the n*(n+1)/2 formula by using the two triangles and a rectangle like this!Or you could just add up how many numbers there are in each row: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
 percentage/706006: is neg 0.763 an irrational number?1 solutions Answer 434962 by jsmallt9(3296)   on 2013-01-24 22:10:59 (Show Source): You can put this solution on YOUR website!A decimal is rational ifit terminates (stops) (like -0.763); orit does not terminate but it has a set of infinitely repeating digits (like 4.8134134134...)A decimal is irrational if it does not terminate and it does not have repeating digits (like or the number "e" or many square roots)
 Polynomials-and-rational-expressions/706033: factor the trinomial 2x^2-5xy-3y^2 1 solutions Answer 434956 by jsmallt9(3296)   on 2013-01-24 22:01:49 (Show Source): You can put this solution on YOUR website!The technique we will use to factor can be described as "un-FOIL-ing". FOIL is used to multiply expressions like (a+b)(c+d). Since factoring is, in effect, "un-multiplying" an expression we can use the reverse of FOIL to turn into an expression of the form (a+b)(c+d). If we have done enough FOIL-ing we know that when (a+b)(c+d) results in a trinomial, like , then:The first term of the trinomial, , usually comes from the "F" part of FOIL (which would be a*c). For there is really only one possible "a" and "c": 2x and x.The third term of the trinomial, , usually comes from the "L" part of FOIL (which would be c*d). For we could have two different c/d pairs: -3y and y or 3y and -y.The middle term of the trinomial, -5xy, usually comes from the sum of the "O" and "I" parts of FOIL (which would be a*d + b*c).Now we just try the different possible values for a, b, c and d to see if any of them will create the proper middle term. (If none do then the trinomial will not factor.) As it turns out, there is a combination that works: a = 2x, b = y, c = x and d = -3y So factors into: (2x + y)(x + (-3y)) or, more simply: (2x + y)(x - 3y) To check this, just use FOIL!
 Polynomials-and-rational-expressions/706029: Factoring by grouping 3x^7=81x^4 1 solutions Answer 434951 by jsmallt9(3296)   on 2013-01-24 21:29:59 (Show Source): You can put this solution on YOUR website! We will use factoring to solve this. But with only two terms, it is pointless (if it is possible at all) to use factoring by grouping to solve this equation. First get one side to be zero (by subtracting the from each side): And then we factor. First, as always, is to factor out the greatest common factor (GCF) if it is not a 1. The GCF here is : Now we try to factor the factor. Among the factoring patterns you learn is the pattern for difference of cubes: . This pattern may be used here since is an obvious cube and 27 is . Using an "x" for the "a" and a "3" for the "b", then pattern tells us how will factor: which simplifies to: With the factoring complete we can now solve. From the Zero Product Property: or or The first two equations are simple to solve. (We should get x = 0 and x = 3.) For the third equation we must use the quadratic formula: which simplifies as follows: With the negative number in the square root we will get only complex number solutions from this equation. If we are not interested in complex number solutions (or if you don't even know about complex numbers) then we stop here and just use the solutions we found from the other two factors: 0 and 3. If we do want complex solutions then we continue: which is short for: or In standard a + bi form this would be: or This makes the solutions, including these complex number solutions: x = 0 or x = 3 or or
 Polynomials-and-rational-expressions/705557: How do I write this square as a trinomial: (3u-1)^21 solutions Answer 434936 by jsmallt9(3296)   on 2013-01-24 21:02:07 (Show Source): You can put this solution on YOUR website!Method 1: One of the patterns you should have learned by now is: This pattern can be used to multiply with the "a" being "3u" and the "b" being "1": (Note the parentheses. Always a good idea when using patterns!) Method 2: means (3u-1)(3u-1). You can use FOIL to multiply this out. (After you multiply and then add like terms, you get the same answer as method 1 .)
 Polynomials-and-rational-expressions/706025: determine p so that 4q+3 is a factor of 20q^3+23q^2-10q+p1 solutions Answer 434933 by jsmallt9(3296)   on 2013-01-24 20:50:54 (Show Source): You can put this solution on YOUR website!If we factor a 4 out of 4q+3 we get So if (4q+3) is a factor of our polynomial then so will . This fact is useful because we can more obviously use synthetic division with the : ```-3/4 | 20 23 -10 p ------ -15 -6 12 -------------------- 20 8 -16 12+p ``` In order for (and therefore (4q+3)) to be a factor, the remainder, 12+p, must be zero. So "p" must be -12.
 Surface-area/705791: Find the area of a label used to cover a can of juice that has a diameter of 13.5 cm and a height of 15 cm. Round to the nearest hundredth.1 solutions Answer 434804 by jsmallt9(3296)   on 2013-01-24 13:41:35 (Show Source): You can put this solution on YOUR website!The problem is to find the lateral surface area of a cylinder. If you know the formula for this you can use it to find the answer. If you do not know the formula it is still possible to find the answer. You just have to realize that the label on the can is simply a rectangle that has been wrapped wound the can. To find the area of this rectangle we just have to know the length and width (or base and height) of the rectangle. The height of the can, 15cm, is one side of the rectangle. The other side of the rectangle is the length of the circular edge at the top (or bottom) of the can. This length is the circumference of the circle. Since the formulas for circumference is: or Since we have the diameter, d, we will use the first formula: or Now that we have the length and width of the rectangle we can find its area: which simplifies to: This is an exact solution for the area of the label. Your problem says to round this to the nearest 100th which I will leave up to you. Just replace the with 3.141592693, multiply by 202.5 and then round.
 Quadratic_Equations/705265: 5x^2-5x=150 found solution using quadratic equation to be 6 and -5. 6^2=36 5*36=180 5*6=30 180-30=150 Use solution in original equation, 6 works, but for the life of me I cannot figure out what I am doing wrong with the -5 solution. -5^2=-25 -25*5=-125 Thanks 5*-5=-25 -125 --25=1001 solutions Answer 434793 by jsmallt9(3296)   on 2013-01-24 13:18:46 (Show Source): You can put this solution on YOUR website!6 and -5 are the correct solutions. When you check -5 as a solution you are supposed to replace the x's in: with a -5. In the first term, the x is being squared. So when we replace it with -5, the -5 should be squared, too. A proper replacement of the x's with 6's and -5's would look like: and Notice the parentheses!! They make little difference with the replacements with 6's. But they are critical when replacing the x's with -5! So it is an extremely good habit to use parentheses when making substitutions. Your mistake was not using the parentheses and then finding -5^2 which means "the negative of 5 squared" which is, of course, -25. What you should have been doing was (-5)^2 which means "the square of -5" which is, of course, 25. With a 25 instead of -25 you will find that -5 does check out.
 Quadratic_Equations/705562: f(x)=-x^2+2x+7 Determine if this has a maximum or a minimum value. Then find the max/min value.1 solutions Answer 434791 by jsmallt9(3296)   on 2013-01-24 13:08:25 (Show Source): You can put this solution on YOUR website! First we should recognize that the graph of a quadratic function, like this one, will have a parabola that opens upward (or downward) as a graph. Such a parabola will have a maximum or minimum value. Since the coefficient of the squared term is negative (the "a") this parabola will open downward. If you picture such a parabola you will realize that the vertex will be the highest (i.e. maximum) value. So f(x) will have a maximum value at its vertex. To find this maximum value we just have to find the vertex of the parabola. This can be done by completing the square to put the equation into vertex form. But it is a little easier if you know that the x coordinate of the vertex of any quadratic function will be: Your "a" is -1 and your "b" is 2. So: which simplifies to: Now to find the maximum value we find the y coordinate of the vertex, f(1): Simplifying... So the vertex of the parabola is (1, 8) and the maximum value for the function is 8.
 Quadratic_Equations/705795: A farmer plans to enclose a rectangular region, using part of his barn for one side and fencing for the other three sides. If the side parallel to the barn is to be twice the length of an adjacent side, and the area of the region is to be 162 ft2, how many feet of fencing should be purchased?1 solutions Answer 434790 by jsmallt9(3296)   on 2013-01-24 12:58:31 (Show Source): You can put this solution on YOUR website!Since the region to be enclosed is rectangular we know its area will be length times width (or base times height). If we call the width "x", then the length will be "2x" (since we are told it is twice the length of the width). We are also told that the area is to be 162 square feet. So the equation we can use is: Now we solve for x. First we simplify: Dividing both sides by 2: Subtracting 81: Factoring: (x+9)(x-9) = 0 Using the Zero Product Property we get: x+9 = 0 or x-9 = 0 Solving these: x = -9 or x = 9 Since x represents the width of the region, we will discard the negative solution. (We are not interested in negatives sides of a region.) This makes 9 the only useable value for x (which is the width). And the length will be twice as much: 18. Finally we answer the question: How much fencing should be purchased? Since the barn will serve as one side of the region we will no need fence for that side. We only need fencing for 1 length and 2 widths: 18 + 2*9 = 18 + 18 = 36 feet of fencing will be required.
 Parallelograms/697587: A parallelogram ABCD has all its sides measure 4, one of the diagonals also measures 4. What is its area? 1 solutions Answer 430132 by jsmallt9(3296)   on 2012-12-31 16:57:09 (Show Source): You can put this solution on YOUR website!The formula for the area of a parallelogram is: A = b*h. The base in these area formulas can be any side of the polygon. Since all the sides of this polygon are 4's then we know our base will be 4. The hard part is the height. If you draw a diagram of what you describe then you will see that an equilateral triangle with sides of 4 is formed by two sides of the parallelogram (rhombus actually) and the diagonal whose length is 4. The height of this triangle will also be the height of the parallelogram. The angles in all equilateral triangles are always 60 degrees. If we draw in the height for the triangle we will have a 30-60-90 right triangle with a hypotenuse of 4. From Trigonometry or from remembering the relationships between the sides of 30-60-90 triangles we should be able to figure out that height is So the area of this parallelogram will be:
 Rational-functions/697590: R(x)=x^4+x^2+5/x^2-361 solutions Answer 430127 by jsmallt9(3296)   on 2012-12-31 16:44:04 (Show Source): You can put this solution on YOUR website!First, what are you/we supposed to do with this?Second, is the function: (which is what you posted) or something else like: If the function is anything other than the first one then put parentheses around the entire numerator and around the entire denominator.
 Pythagorean-theorem/697554: the length of one leg of a right triangle is 17cm more than that of the other leg. The length of the hypotenue is 4 vm more than triple that of the shorter leg. Find the lenghts of each of the three sides1 solutions Answer 430125 by jsmallt9(3296)   on 2012-12-31 16:36:21 (Show Source): You can put this solution on YOUR website!Let x = the shorter leg. Then the longer leg would be (x + 17) and the hypotenuse would be (3x + 4). In order for this to be a right triangle these sides must fit the Pythagorean equation: Now we solve for x. First we simplify. (Be sure to use FOIL or the pattern to square the x+17 and the 3x+4.) Since this is a quadratic equation we want one side to be zero. Subtracting the entire left side from each side we get: This will factor, but not easily. So you may prefer to use the Quadratic Formula instead. From the Zero Product Property: 7x + 39 = 0 or x - 7 = 0 Solving these we get: x = -39/7 or x = 7 We will reject the first solution not because it is a fraction but because it is negative. x is the length of the shorter leg and we cannot have negative lengths. So the shorter leg is 7 cm. The longer leg would be x + 17 = 7 + 17 = 24 cm And the hypotenuse would be 3x + 4 = 3(7) + 4 = 21 + 4 = 25 cm
 Triangles/697592: in right triangle ABC, angle C=90, angle A=55, and CA=10. what is the length of AB to the nearest integer. 1 solutions Answer 430123 by jsmallt9(3296)   on 2012-12-31 16:22:44 (Show Source): You can put this solution on YOUR website!Since side CA is adjacent to angle A and side AB is the hypotenuse. The Trig functions that have adjacent and hypotenuse in their ratios are cos and sec. Since our calculators probably do not have a sec button we will use cos: where x is the length of side AB Multiplying each side be x we can eliminate the fraction: Dividing each side by cos(55): This is an exact expression for the solution. For a decimal approximation we use our calculators to divide 10 by cos(55): Rounded to the nearest integer this would be 17.
 Inequalities/695832: What is the solution of the system of inequalities? y≥ x^2+2x+2 y< -x^2-4x-2 Thank You!1 solutions Answer 430120 by jsmallt9(3296)   on 2012-12-31 16:13:53 (Show Source): You can put this solution on YOUR website!First let's look at the graph of the two parabolas: The red one is and the green one is . The solution to would be the red parabola and all the area above/inside the bowl. The solution to would be all the area below/inside the green parabola. (Not the green parabola itself because that is where y equals and the inequality does not include "or equal to".) So the solution to the system is where these two areas overlap each other: The enclosed area between the two parabolas. To express this solution we must first find the two points where the parabolas intersect. Setting and solving for x we should be able to find the points of intersection. Subtracting the entire right side we get: Factor out the GCF of 2: Factoring the trinomial: From the Zero Product Property: x + 1 = 0 or x + 2 = 0 Solving: x = -1 or x = -2 Using these x values and either equation for the parabola we can find that the y values for each x are 1 and 2, respectively. So the points of intersection are: (-1, 1) and (-2, 2) So the x values of the solution are between -1 and -2, inclusive: and the y values of the solution are:
 Quadratic_Equations/697489: Find the complex zeros of the polynomial function. Write f in factored form. Use the complex zeros to write f in factored form.1 solutions Answer 430078 by jsmallt9(3296)   on 2012-12-31 07:59:15 (Show Source): You can put this solution on YOUR website! The problem tells us nothing about any zeros or factors so we will have to find them on our own. Always start factoring by factoring our the greatest common factor (unless it is a 1 which we rarely bother factoring out). The GCF is a 1 so we will not factor it out. Next we usually try factoring by patterns or trinomial factoring because they are often easier than the other two factoring techniques. However out expression has too many terms for a trinomial or for any of the patterns. Now we are left with factoring by grouping or with factoring by trial and error of the possible rational roots. Factoring by grouping requires an even number of terms. Since we have 5 terms we would have to split the into two parts before we try to use factoring by grouping. Since I do not see which particular split (, , , etc.) I am going to try to avoid using this method an go on to factoring by trial and error of the possible rational roots. The possible rational roots of a polynomial are all the ratios, positive and negative, which can be formed by a factor of the constant term (at the end) over a factor of the leading coefficient (at the beginning). Our constant term is 64 and our leading coefficient is 3. The factors of 64 are 1, 2, 4, 8, 16, 32, 64 and the factors of 3 are just 1 and 3. So the possible rational roots of f are: +1/1 (or +1), +2/1 (or +2), +4/1 (or +4), +8/1 (or +8), +16/1 (or +16), +32/1 (or +32), +64/1 (or +64), +1/3, +2/3, +4/3, +8/3, +16/3, +32/3, +64/3 With so many possible rational roots it could take quite a while to find some roots. It will help if we use our brains to eliminate some of the possible roots logically. With only two terms with negative coefficients, it is unlikely the larger positive roots could make f be zero. So we will not try 4, 8, 16, 32 or 64 (unless nothing else works). Let's try 2 (using synthetic division which I hope you have learned): ```2 | 3 -19 54 176 -64 ---- 6 -26 56 264 -------------------------- 3 -13 28 232 200 ``` The 200 in the lower right corner is the remainder. The remainder is f(2). Since f(2) is not zero, 2 is not a root. (And the fact that it is a large positive number tells us that even 2 was too large of a positive root to work.) We can try 1 and -1 mentally since powers of 1 and -1 are easy: f(1) = 3 - 19 + 54 + 176 -64 = 150. f(-1) = 3 + 19 + 54 - 176 -64 = -164 Neither of them work. But the fact that f(1) is positive and f(-1) is negative tells us that there is a root between them. Since zero is nearly halfway between 150 and -164 and since 150 is a little closer to zero, I'm going to guess that 1/3 might be a root. Let's see: ```1/3 | 3 -19 54 176 -64 ---- 1 -6 16 64 -------------------------- 3 -18 48 192 0 ``` Bingo! The remainder is zero so f(1/3) = 0 and (x - 1/3) is a factor. Not only that, the rest of the bottom line is the other factor of f. "3 -18 48 192" translates into: . So: We can simplify this a little if we factor out a 3 from the second factor: and then multiply (x - 1/3) by the 3: Now we continue to factor. The last factor, , has the same constant term, 64, but the leading coefficient is now a 1. So we have the same list of possible rational roots as before except for the fractions. And since a rational root that didn't work before (-1, 1, 2, 4, 8, 16. 32 and 64) cannot magically start working later we only have -2, -4, -8, -16, -32 and -64 left to try. Let's start with -2: ```-2 | 1 -6 16 64 ---- -2 16 -64 ------------------ 1 -8 32 0 ``` The remainder is zero. So f(-2) = 0 and (x - (-2)) or (x + 2) is a factor. And the other factor is : The last factor is a quadratic. So we can use alternate factoring techniques. But no matter what technique we use, it will not factor. But we can use the quadratic formula to find its roots: which simplifies as follows: which is short for: or With these two roots we can write the remaining factors: (x - (4 + 4i)) and (x - (4 - 4i)) which simplify to: (x - 4 - 4i)) and (x - 4 + 4i)) Adding these two factors to our factored f:
 logarithm/697486: Expand using the log rules my work: I'm not sure if I used the log rules correctly. Is this the answer or am i missing some steps to expanding the log?1 solutions Answer 430075 by jsmallt9(3296)   on 2012-12-31 06:48:50 (Show Source): You can put this solution on YOUR website!First of all, please try to post your questions in an appropriate category. You will get a quicker response if you do so. This problem has nothing to do with quadratic equations. I have changed it to an appropriate category. Your work is all good so far. And you may actually be finished. It depends on the exact meaning of "expand". For many teachers they want you to make the arguments as simple as possible. If this is the case then you need to use one more rule, , to move the exponent of the "c" out in front:
 logarithm/697488: solve for x: log(base of x)27=3 my work: x=3 I'm not sure if I solved the log correctly.1 solutions Answer 430074 by jsmallt9(3296)   on 2012-12-31 06:18:01 (Show Source): You can put this solution on YOUR website!100% correct!