💡First, I need to evaluate the expression (frac{7 + x(4 + x) - 4^2}{x - 4 + x^2}) for (x = -3).I'll start by substituting (x = -3) into the expression:[frac{7 + (-3)(4 + (-3)) - 4^2}{-3 - 4 + (-3)^2}]Next, I'll simplify the numerator:[7 + (-3)(1) - 16 = 7 - 3 - 16 = -12]Then, I'll simplify the denominator:[-3 - 4 + 9 = 2]Finally, I'll divide the simplified numerator by the simplified denominator:[frac{-12}{2} = -6]So, the final answer is (-6).

💡First, I recognize that the standard deviation of the original set of numbers ( a_1, a_2, a_3, a_4, a_5 ) is 2. Since variance is the square of the standard deviation, the variance of the original set is ( 2^2 = 4 ).Next, I consider the transformation applied to each number in the set: multiplying by 3 and then subtracting 2. Adding a constant (in this case, subtracting 2) does not affect the variance because it shifts all values uniformly without changing their spread. However, multiplying each value by a constant (here, 3) scales the variance by the square of that constant.Therefore, the variance of the transformed set ( 3a_1 - 2, 3a_2 - 2, 3a_3 - 2, 3a_4 - 2, 3a_5 - 2 ) is calculated by multiplying the original variance by ( 3^2 ), which is 9. This results in a variance of ( 4 times 9 = 36 ).

💡First, I need to determine the cost per Douglas fir tree that Sawz Logging Co. paid.I know that the company bought a total of 850 trees, with 350 being Douglas fir trees. This means the remaining 500 trees are ponderosa pine trees.The cost for each ponderosa pine tree is 225, so the total cost for the ponderosa pine trees is 500 multiplied by 225, which equals 112,500.The total amount paid for all the trees is 217,500. To find the total cost for the Douglas fir trees, I subtract the cost of the ponderosa pine trees from the total amount: 217,500 minus 112,500 equals 105,000.Now, to find the cost per Douglas fir tree, I divide the total cost for Douglas fir trees by the number of Douglas fir trees: 105,000 divided by 350 equals 300.Therefore, Sawz Logging Co. paid 300 for each Douglas fir tree.

💡Okay, so I have this problem here where I need to find out how many mappings there are from set A to set B. Set A is {-1, 0, 1} and set B is {a, b}. Hmm, let me try to understand what a mapping is first. From what I remember, a mapping, or a function, from set A to set B assigns each element in A to exactly one element in B. So, for every element in A, I have to choose an element in B to map it to. Alright, so set A has three elements: -1, 0, and 1. Set B has two elements: a and b. That means for each element in A, I have two choices in B. Let me break it down. For the first element in A, which is -1, I can map it to either a or b. So that's two options. Then, for the next element, which is 0, I can also map it to either a or b. Again, two options. And finally, for the element 1, I can map it to a or b as well. So, if I think about it, for each of the three elements in A, there are two choices in B. That seems like a multiplication problem. Like, for each element, I multiply the number of choices together. So, 2 choices for -1, times 2 choices for 0, times 2 choices for 1. Let me write that out: 2 × 2 × 2. That equals 8. So, does that mean there are 8 possible mappings from A to B? Wait, let me make sure I'm not missing something. Is there any restriction on the mappings? Like, do I have to map each element uniquely, or can multiple elements map to the same element in B? I think in a general mapping, multiple elements can map to the same element in B. So, for example, both -1 and 0 could map to a, and 1 maps to b. That's still a valid mapping. So, the fact that multiple elements can map to the same element doesn't restrict the number of choices; it actually allows for more possibilities. So, going back, since each element in A has two independent choices in B, and there are three elements, the total number of mappings is indeed 2^3, which is 8. Let me try to list them out to double-check. 1. -1 → a, 0 → a, 1 → a2. -1 → a, 0 → a, 1 → b3. -1 → a, 0 → b, 1 → a4. -1 → a, 0 → b, 1 → b5. -1 → b, 0 → a, 1 → a6. -1 → b, 0 → a, 1 → b7. -1 → b, 0 → b, 1 → a8. -1 → b, 0 → b, 1 → bYep, that's eight different mappings. Each one is unique because at least one element in A maps differently. I think that makes sense. So, the number of mappings from A to B is 8.

💡First, I need to determine the total number of participants in the meeting, given the information about the number of female and male democrats and the overall proportion of democrats.I'll start by defining the variables:- Let ( F ) represent the number of female participants.- Let ( M ) represent the number of male participants.From the problem, I know that half of the female participants are democrats, so the number of female democrats is ( frac{F}{2} ). It's also given that there are 130 female democrats, which means:[frac{F}{2} = 130]Solving for ( F ):[F = 260]Next, one-quarter of the male participants are democrats, so the number of male democrats is ( frac{M}{4} ).The total number of democrats is the sum of female and male democrats:[frac{F}{2} + frac{M}{4}]It's also given that one-third of all participants are democrats:[frac{F + M}{3}]Setting these equal to each other:[frac{F}{2} + frac{M}{4} = frac{F + M}{3}]Substituting ( F = 260 ) into the equation:[frac{260}{2} + frac{M}{4} = frac{260 + M}{3}]Simplifying:[130 + frac{M}{4} = frac{260 + M}{3}]To eliminate the fractions, I'll multiply the entire equation by 12 (the least common multiple of 2, 4, and 3):[12 times 130 + 3M = 4 times (260 + M)]Calculating:[1560 + 3M = 1040 + 4M]Subtracting ( 3M ) from both sides:[1560 = 1040 + M]Solving for ( M ):[M = 1560 - 1040 = 520]Now, I have the number of male participants (( M = 520 )) and the number of female participants (( F = 260 )). The total number of participants is:[F + M = 260 + 520 = 780]

💡Okay, so I have this problem here: If x and y are positive real numbers, and they satisfy the equation x + 4y = 1, then what is the maximum value of the product x times y? Hmm, let me try to figure this out step by step.First, I know that x and y are positive, so both x and y are greater than zero. The equation given is x + 4y = 1, which is a linear equation. I need to maximize the product xy. This seems like an optimization problem with a constraint.I remember that for optimization problems with constraints, one common method is to use substitution. Since we have two variables, x and y, and one equation, maybe I can express one variable in terms of the other and then substitute it into the product formula.Let me try that. From the equation x + 4y = 1, I can solve for x:x = 1 - 4yNow, substitute this into the product xy:xy = (1 - 4y)y = y - 4y²So, now the product is expressed as a function of y: f(y) = y - 4y². This is a quadratic function, and since the coefficient of y² is negative (-4), the parabola opens downward, meaning the vertex is the maximum point.To find the maximum value, I can find the vertex of this parabola. The general form of a quadratic function is f(y) = ay² + by + c, and the vertex occurs at y = -b/(2a). In this case, a = -4 and b = 1.So, plugging in the values:y = -b/(2a) = -1/(2*(-4)) = -1/(-8) = 1/8So, the value of y that maximizes the product is 1/8. Now, let's find the corresponding x value using the equation x = 1 - 4y:x = 1 - 4*(1/8) = 1 - 0.5 = 0.5So, x is 0.5 when y is 1/8. Now, let's compute the product xy:xy = (0.5)*(1/8) = 0.5/8 = 1/16Therefore, the maximum value of xy is 1/16.Wait, let me double-check my steps to make sure I didn't make any mistakes. I started by expressing x in terms of y, substituted it into the product, got a quadratic function, found the vertex, and then computed the product. Everything seems to add up.Alternatively, I remember there's a method using the AM-GM inequality, which might be applicable here. The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Maybe I can apply that here.Let me see. The equation is x + 4y = 1. I can think of this as two terms: x and 4y. So, the arithmetic mean of x and 4y is (x + 4y)/2 = 1/2. The geometric mean is sqrt(x * 4y) = sqrt(4xy) = 2*sqrt(xy).According to the AM-GM inequality:(x + 4y)/2 ≥ sqrt(x * 4y)Which simplifies to:1/2 ≥ 2*sqrt(xy)Divide both sides by 2:1/4 ≥ sqrt(xy)Square both sides:1/16 ≥ xySo, this tells me that the maximum value of xy is 1/16, which matches what I found earlier. That's a good confirmation.Also, equality in AM-GM holds when all the terms are equal. So, in this case, x must be equal to 4y. From x = 4y, and x + 4y = 1, we can substitute:4y + 4y = 1 => 8y = 1 => y = 1/8Then, x = 4y = 4*(1/8) = 1/2, which again gives us xy = (1/2)*(1/8) = 1/16.So, both methods give me the same result, which makes me more confident that 1/16 is indeed the correct maximum value.I think I've covered all the necessary steps and double-checked my work using two different methods. It seems solid.**Final Answer**The maximum value of ( x cdot y ) is boxed{dfrac{1}{16}}.