💡First, determine the number of gold bars Legacy has. She has 5 bars.Next, calculate the total value of Legacy's gold by multiplying the number of bars by the value per bar:5 bars * 2200/bar = 11000Aleena has 2 bars fewer than Legacy, so she has:5 bars - 2 bars = 3 barsCalculate the total value of Aleena's gold:3 bars * 2200/bar = 6600Finally, add the total values of both Legacy's and Aleena's gold to find the combined total value:11000 + 6600 = 17600

💡First, I need to determine the total number of students across all five vans by adding the number of students in each van.Next, I will subtract the total number of boys from the total number of students to find out how many girls there are.Finally, I will present the calculated number of girls as the answer.

💡First, I need to identify which numbers in the given list are non-negative. Non-negative numbers are those that are greater than or equal to zero.Looking at each number in the list:- (-8) is negative.- (2.1) is positive.- (dfrac{1}{9}) is positive.- (3) is positive.- (0) is non-negative.- (-2.5) is negative.- (10) is positive.- (-1) is negative.Counting the non-negative numbers:- (2.1)- (dfrac{1}{9})- (3)- (0)- (10)There are a total of 5 non-negative numbers.

💡Okay, so I have this problem where points A and B are 6 units apart. I need to figure out how many lines in the plane containing A and B are 3 units away from both A and B. The answer choices are from 0 to more than 3, so I need to visualize this.First, let me recall that the set of all points that are 3 units from A forms a circle with center A and radius 3. Similarly, the set of all points that are 3 units from B forms another circle with center B and radius 3. So, I can imagine two circles, each with radius 3, and the distance between their centers, A and B, is 6 units.Now, if I think about the distance between the centers and the radii, I remember that if the distance between centers is equal to the sum of the radii, the circles are externally tangent. In this case, the distance between A and B is 6, and the sum of the radii is 3 + 3 = 6. So, the circles are externally tangent to each other. That means they touch at exactly one point.Next, I need to figure out how many lines can be tangent to both circles. When two circles are externally tangent, there are three types of common tangents: two external tangents and one common tangent at the point where they touch. Wait, no, actually, when two circles are externally tangent, they have three common tangents: two external ones that don't pass through the point of contact, and one common tangent at the point where they touch. So, that's three lines in total.But hold on, the question is about lines that are 3 units from both A and B. Each such line must be a common tangent to both circles. Since the circles are externally tangent, there are three common tangents. Therefore, there should be three lines that satisfy the condition.Wait, let me double-check. If the circles were separate, not touching, they would have four common tangents: two external and two internal. But since they are externally tangent, the two internal tangents coincide at the point of contact, so we only have three distinct common tangents. That makes sense.So, putting it all together, the number of lines that are 3 units from both A and B is 3.**Final Answer**boxed{D}

💡Okay, so I have this problem here about a circle and two points, A and B. The circle is given by the equation (x - 3)² + (y - 4)² = 1. Points A and B are at (-1, 0) and (1, 0) respectively. I need to find the maximum and minimum values of d, where d is defined as |PA|² + |PB|², and P is a moving point on the circle.First, I should probably write down what |PA|² and |PB|² mean. Since P is a point (x, y), then |PA|² is the square of the distance between P and A, which would be (x - (-1))² + (y - 0)², right? So that simplifies to (x + 1)² + y². Similarly, |PB|² would be (x - 1)² + y².So, d = |PA|² + |PB|² = (x + 1)² + y² + (x - 1)² + y². Let me expand these terms:(x + 1)² = x² + 2x + 1(x - 1)² = x² - 2x + 1Adding these together with the y² terms:d = (x² + 2x + 1 + y²) + (x² - 2x + 1 + y²)Combine like terms:x² + x² = 2x²2x - 2x = 0 (they cancel out)1 + 1 = 2y² + y² = 2y²So, d = 2x² + 2y² + 2Hmm, that's interesting. So, d simplifies to 2(x² + y²) + 2. That means if I can find the maximum and minimum values of x² + y² on the circle, I can plug them into this equation to find d's maximum and minimum.Wait, x² + y² is the square of the distance from the origin to the point (x, y). So, if I can find the maximum and minimum distances from the origin to points on the circle C, then square them, I can find the maximum and minimum of x² + y².But how do I find the maximum and minimum distances from the origin to the circle C?The circle C is centered at (3, 4) with a radius of 1. The origin is at (0, 0). So, the distance from the origin to the center of the circle is sqrt[(3 - 0)² + (4 - 0)²] = sqrt[9 + 16] = sqrt[25] = 5.Since the radius of the circle is 1, the maximum distance from the origin to any point on the circle would be the distance from the origin to the center plus the radius, which is 5 + 1 = 6. Similarly, the minimum distance would be the distance from the origin to the center minus the radius, which is 5 - 1 = 4.Therefore, the maximum value of x² + y² is 6² = 36, and the minimum is 4² = 16.Plugging these back into the expression for d:Maximum d = 2*(36) + 2 = 72 + 2 = 74Minimum d = 2*(16) + 2 = 32 + 2 = 34So, the maximum value of d is 74, and the minimum is 34.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I started by expanding |PA|² and |PB|² correctly.2. I combined the terms correctly, resulting in d = 2x² + 2y² + 2.3. I recognized that x² + y² is the square of the distance from the origin.4. Calculated the distance from the origin to the center of the circle as 5.5. Added and subtracted the radius to find the max and min distances: 6 and 4.6. Squared them to get 36 and 16.7. Plugged into d to get 74 and 34.Everything seems to check out. I don't think I made any errors in the calculations.Alternatively, another way to approach this might be to use coordinates relative to the center of the circle. Let me try that method to confirm.Let’s denote the center of the circle as point O(3, 4). Any point P on the circle can be expressed as O plus a vector of length 1 in some direction. So, P = (3 + cosθ, 4 + sinθ), where θ is the angle parameter.Then, let's compute |PA|² and |PB|².First, |PA|² = (3 + cosθ + 1)² + (4 + sinθ - 0)² = (4 + cosθ)² + (4 + sinθ)²Similarly, |PB|² = (3 + cosθ - 1)² + (4 + sinθ - 0)² = (2 + cosθ)² + (4 + sinθ)²So, d = |PA|² + |PB|² = [(4 + cosθ)² + (4 + sinθ)²] + [(2 + cosθ)² + (4 + sinθ)²]Let me expand each term:First term: (4 + cosθ)² = 16 + 8cosθ + cos²θSecond term: (4 + sinθ)² = 16 + 8sinθ + sin²θThird term: (2 + cosθ)² = 4 + 4cosθ + cos²θFourth term: (4 + sinθ)² = 16 + 8sinθ + sin²θNow, adding all these together:First term + Second term + Third term + Fourth term= [16 + 8cosθ + cos²θ] + [16 + 8sinθ + sin²θ] + [4 + 4cosθ + cos²θ] + [16 + 8sinθ + sin²θ]Combine like terms:Constants: 16 + 16 + 4 + 16 = 52Cosine terms: 8cosθ + 4cosθ = 12cosθSine terms: 8sinθ + 8sinθ = 16sinθCos²θ terms: cos²θ + cos²θ = 2cos²θSin²θ terms: sin²θ + sin²θ = 2sin²θSo, d = 52 + 12cosθ + 16sinθ + 2cos²θ + 2sin²θBut wait, cos²θ + sin²θ = 1, so 2cos²θ + 2sin²θ = 2*1 = 2Therefore, d = 52 + 12cosθ + 16sinθ + 2 = 54 + 12cosθ + 16sinθHmm, so now d is expressed as 54 + 12cosθ + 16sinθ. To find the maximum and minimum of this expression, I can treat it as a function of θ.The expression 12cosθ + 16sinθ can be written in the form Rcos(θ - φ), where R is the amplitude and φ is the phase shift.Calculating R: R = sqrt(12² + 16²) = sqrt(144 + 256) = sqrt(400) = 20So, 12cosθ + 16sinθ = 20cos(θ - φ), where φ is such that cosφ = 12/20 = 3/5 and sinφ = 16/20 = 4/5.Therefore, the maximum value of 12cosθ + 16sinθ is 20, and the minimum is -20.Thus, d = 54 + 20cos(θ - φ). The maximum value of d is 54 + 20 = 74, and the minimum is 54 - 20 = 34.So, this method also gives the same result: maximum d is 74, minimum d is 34.Therefore, both methods confirm that the maximum and minimum values of d are 74 and 34 respectively.I think this solidifies the answer. I don't see any mistakes in either approach, so I feel confident that 74 and 34 are the correct maximum and minimum values.**Final Answer**The maximum value of ( d ) is (boxed{74}) and the minimum value is (boxed{34}).

💡Okay, so I've got this problem here about cutting a piece of paper into 5 pieces, then taking some of those pieces and cutting each of them into 5 again, and repeating this process until I stop. The question is asking what the possible total number of pieces could be, with the options being 1990, 1991, 1992, or 1993.Alright, let me try to break this down. First, when I cut a piece of paper into 5 pieces, I'm increasing the number of pieces by 4, right? Because 1 piece becomes 5, so that's an increase of 4. So, every time I cut a piece into 5, I add 4 more pieces.Now, if I start with 1 piece and cut it into 5, I have 5 pieces. Then, if I take, say, 1 of those 5 pieces and cut it into 5 again, I'll have 5 + 4 = 9 pieces. Wait, no, that's not quite right. If I take 1 piece and cut it into 5, I'm replacing 1 piece with 5, so the total number of pieces becomes 5 - 1 + 5 = 9. Yeah, that makes sense.So, each time I cut a piece into 5, I add 4 pieces. Therefore, the total number of pieces after each cut increases by 4. So, starting from 1, then 5, then 9, then 13, and so on. It seems like the number of pieces follows an arithmetic sequence where each term increases by 4.Wait, but hold on. The problem says "take a certain number and cut each of them into 5 pieces again." So, it's not necessarily just cutting one piece each time. I could cut multiple pieces each time, right? So, if I have, say, 5 pieces and I decide to cut all 5 of them into 5 pieces each, then each of those 5 pieces becomes 5, so I'd have 5 * 5 = 25 pieces. But that seems like a big jump.But in the problem, it says "take a certain number and cut each of them into 5 pieces again." So, it's not specified whether I have to cut all of them or just some. So, I guess I can choose any number of pieces to cut each time, from 1 up to the total number of pieces I have.So, if I have N pieces, I can choose to cut K pieces, where K is between 1 and N, and each of those K pieces will become 5 pieces. So, the total number of pieces after that operation would be N - K + 5K = N + 4K.So, each time I cut K pieces, the total number of pieces increases by 4K.Therefore, the total number of pieces after each operation is always going to be 1 + 4 times the total number of cuts made across all pieces.Wait, let me think about that again. If I start with 1 piece, and each time I cut K pieces, each cut adds 4 pieces. So, the total number of pieces is 1 + 4 times the number of cuts.But actually, each cut operation on a piece adds 4 pieces, so if I make M cuts in total, the total number of pieces is 1 + 4M.But M is the total number of pieces I've cut. So, for example, if I cut 1 piece once, M=1, total pieces=5. If I cut 1 piece again, M=2, total pieces=9. If I cut 2 pieces, M=2, total pieces=1 + 4*2=9 as well.Wait, that doesn't seem right. If I have 5 pieces and I cut 2 of them, each cut adds 4 pieces, so total pieces would be 5 - 2 + 2*5 = 5 - 2 + 10 = 13. Which is 1 + 4*3=13, because M=3 cuts? Wait, no, M=2 cuts.Hmm, maybe my initial thought was wrong. Let me try to formalize this.Let’s denote the number of pieces as P. Initially, P=1.Each time I cut K pieces, each of those K pieces becomes 5, so the number of pieces becomes P - K + 5K = P + 4K.So, the total number of pieces increases by 4K each time I cut K pieces.Therefore, the total number of pieces after several operations would be 1 + 4*(K1 + K2 + K3 + ... + Kn), where K1, K2, etc., are the number of pieces cut in each operation.So, the total number of pieces is always of the form 1 + 4*M, where M is the total number of pieces cut across all operations.Therefore, the total number of pieces must be congruent to 1 modulo 4.So, P ≡ 1 mod 4.Now, looking at the options:1990: 1990 divided by 4 is 497.5, so remainder 2.1991: 1991 divided by 4 is 497.75, so remainder 3.1992: 1992 divided by 4 is 498, so remainder 0.1993: 1993 divided by 4 is 498.25, so remainder 1.Therefore, 1993 is the only number that is congruent to 1 modulo 4 among the options.So, the possible total number of pieces is 1993.