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Let x = the amount of 4% (0.04) solution that is mixed with 6 gallons of 9% (0.09) solution to get some quantity of 6% solution (0.06).  We can set up our equation by multiplying the percentage of the solution to the volume: 0.04x + (0.09) (8) = (0.06)(x + 8) simplify and...

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In Scientific notation, you only have one digit to the left of decimal, and the rest of the digits are to the right, leaving out any unnecessary zeros. This is then multiplied by some power of 10, which can be determined by the number of places the decimal place moves.  Given 53 030...

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Assuming the pole was perpendicular to the ground when the woodpecker arrived, picture a right triangle. Since the top was 10 m from the base, we know that one leg was 10 m, which means that the other leg plus the hypotenuse added to 19. We can express the leg as x and the hypotenuse as 19 -...

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The domain values are those within the parentheses. The range values are the result.  So the domain is {a, b, c, d} and the range is {p, r, t} The ordered pairs are (a,r) (b, p) (c, t) (d, r)

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When finding the LCD, it is often helpful to factor the denominators, then see what you have: In the first problem, our denominators are x2 - 12x + 35     and x - 7 x2 - 12x + 35  factors into (x - 5)(x - 7).  So your LCD is (x - 5)(x - 7) So, multiply...

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Factors of 25 are 1 * 25, 5 * 5 Factors of 12 are 1 * 12, 2 * 6, 3 * 4 Since we're looking for two values that, when the  difference of the products is 5, I'm thinking we need values close together... So I'm going to look at 5*5 and 3 * 4: (3x + 5)(4x - 5) Since the constant...

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Cos (7 pi/ 12) (answer)

Try using the half angle formula: cos (x/2) = +/- √[(1 + cos x)/2] since 7pi/12 = 1/2 (7 pi/6),  cos (7pi/12) = +/- √[(1 + cos (7pi/6))/2] I'll leave the rest to the student.

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From trignometric identities, cos 2x = cos 2x - sin2x  and sin 2x = 2 sin x cos x. We can replace these as  cos 2x / [(sin 2x)/2] and simplify to  (2 cos 2x)/(sin 2x) (Since cot x = cos x/sin x, you could also write this as 2 cot 2x)

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For the first two, find a common factor. In the first, pull a 7 out: 7(d2 - 2d - 3) = 0 . Now factor and solve.  For the third, distribute across the parentheses, gather like terms, and set equal to zero. If there are any common factors, pull them out and factor what is left: 6c(c...

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I've worked this 2 ways, and get close, but it's not  the expected result. I'm posting in hopes that it will put you  (or one of my esteemed colleagues) on the right track: The identity for sin 3Θ =  3 sin Θ - 4 sin3 Θ. So if we replace this in the original equation,...

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It would depend on the type of pyramid.  If it was equilateral (that is, all edges were the same), it wouldn't matter - something like identifying the base of a cube.  (Trivia - this would also be called a tetrahedron) If it was isoceles - that is, the edges of faces were the same...

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There is not a hands-down, "best way" - it depends on the job you are trying to do, just like there is no "best hammer" in a handyman's toolbox.  The two most common ways are elimination and substitution (I'll spare describing matrices).  In Elimination, you...

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Square each to get x2/a2 cos2 Θ + xy/ab cos Θ sin Θ + y2/b2 sin 2 Θ = 1 and  x2/a2 sin2 Θ - xy/ab cos Θ sin Θ + y2/b2 cos 2 Θ = 1. Now add them together - note that 1 - your center terms drop out  2 - factor common terms and you...

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sin 2Θ = 2 sin Θ cos Θ so then 2 sin Θ cos Θ = cos Θ Divide both sides by cos Θ  to get 2 sin Θ = 1 sin Θ = 1/2 So where does sin Θ = 1/2 ? That is left for the student to determine (hint: there are two values)

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cos (A - B) = cos A cos B + sin A sin B cos (A + B) = cos A cos B - sin A sin B so  cos (A - B) - cos (A + B) = [cos A cos B + sin A sin B] - [cos A cos B - sin A sin B] Simplify and solve.  

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A line that is perpendicular will have a slope that is the negative reciprocal of the slope of a given line. For example, if the slope of a line was 3, then the perpendicular line would have slope -1/3 Given x - y = 3, let's get this into slope-intercept form y = mx + b -y = -x +...

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Let t = the number of years since 1992. Let k = the linear coefficient.  In 1992, we can represent the population as P=4620 + kt = 4620 + k(0) = 4620 (but we knew that). We can then use P=4620 + kt for our 1996 population: P=4620 + k(4) = 4020 4k = 4020 - 4620...

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Margarita - An angle usually represents a measurement - such as 90 degrees or 1/2 pi radians.  A vertex generally represents the "point" of the angle (or in later cases, a parabola or an ellipse) where the direction changes. Let's assume you were walking on the lower edge...

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