answer：First, I'll define the variables. Let ( x ) represent the number of 12-glass boxes and ( y ) represent the number of 16-glass boxes.According to the problem, there are 16 more 16-glass boxes than 12-glass boxes. This gives me the equation:[y = x + 16]Next, the average number of glasses per box is 15. This average is calculated by dividing the total number of glasses by the total number of boxes. So, I can write the equation for the average as:[frac{12x + 16y}{x + y} = 15]Now, I'll substitute ( y ) from the first equation into the second equation:[frac{12x + 16(x + 16)}{x + (x + 16)} = 15]Simplifying the equation:[frac{12x + 16x + 256}{2x + 16} = 15][frac{28x + 256}{2x + 16} = 15]Multiplying both sides by ( 2x + 16 ) to eliminate the denominator:[28x + 256 = 15(2x + 16)][28x + 256 = 30x + 240]Subtracting ( 28x ) from both sides:[256 = 2x + 240]Subtracting 240 from both sides:[16 = 2x]Dividing by 2:[x = 8]Now that I have ( x = 8 ), I can find ( y ) by substituting ( x ) back into the first equation:[y = x + 16][y = 8 + 16][y = 24]Finally, I'll calculate the total number of glasses:[text{Total glasses} = (12text{-glass boxes} times 12 text{ glasses/box}) + (16text{-glass boxes} times 16 text{ glasses/box})][text{Total glasses} = (8 times 12) + (24 times 16)][text{Total glasses} = 96 + 384][text{Total glasses} = 480]

answer：Okay, so I need to figure out how many five-digit integers are palindromes. Hmm, palindromes are numbers that read the same forwards and backwards, right? So, for a five-digit number, the first digit has to be the same as the fifth digit, and the second digit has to be the same as the fourth digit. The middle digit can be anything.Let me break it down. A five-digit palindrome would look like this: ABCBA. Here, A, B, and C are digits. But wait, A can't be zero because it's the first digit of a five-digit number. So, A has to be from 1 to 9. B and C, on the other hand, can be any digit from 0 to 9 because they can be in the middle or the second position.So, let's think about how many choices we have for each letter. For A, since it can't be zero, there are 9 possible choices (1 through 9). For B, since it can be any digit, there are 10 choices (0 through 9). Similarly, for C, there are also 10 choices.Now, since the number is ABCBA, once we choose A, B, and C, the rest of the digits are determined. That means the total number of five-digit palindromes is the product of the number of choices for each of these letters.So, the total number should be 9 (choices for A) multiplied by 10 (choices for B) multiplied by 10 (choices for C). Let me write that out:Total palindromes = 9 × 10 × 10 = 900.Wait, does that make sense? Let me double-check. If A has 9 options, B has 10, and C has 10, then yes, multiplying them together gives the total combinations. So, 9 × 10 is 90, and 90 × 10 is 900. That seems right.I don't think I'm missing anything here. Each digit's choice is independent of the others, except for the constraints on A. So, the calculation should be straightforward. Yeah, I think 900 is the correct number of five-digit palindromes.

answer：Okay, so I have this problem where I need to find a complex number z and the ratio of z to its conjugate, given that (1 + 2i) times the conjugate of z equals 4 + 3i. Hmm, let me try to break this down step by step.First, I remember that for any complex number z, its conjugate, denoted as (overline{z}), is obtained by changing the sign of the imaginary part. So if z is a + bi, then (overline{z}) is a - bi. That makes sense. The equation given is ((1 + 2i)overline{z} = 4 + 3i). I need to solve for (overline{z}) first, and then find z from that. To isolate (overline{z}), I should divide both sides of the equation by (1 + 2i). So, (overline{z} = frac{4 + 3i}{1 + 2i}).But dividing complex numbers can be tricky. I recall that to simplify such expressions, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (1 + 2i) is (1 - 2i). So, let me do that:[overline{z} = frac{4 + 3i}{1 + 2i} times frac{1 - 2i}{1 - 2i}]Multiplying out the numerators and denominators:Numerator: (4 + 3i)(1 - 2i)Denominator: (1 + 2i)(1 - 2i)Let me compute the numerator first:4*1 = 44*(-2i) = -8i3i*1 = 3i3i*(-2i) = -6i²So, adding those up: 4 - 8i + 3i - 6i²I remember that i² = -1, so -6i² becomes +6.So, combining like terms: 4 + 6 = 10, and -8i + 3i = -5i.So, numerator is 10 - 5i.Now, the denominator:(1 + 2i)(1 - 2i) = 1*1 + 1*(-2i) + 2i*1 + 2i*(-2i)Which is 1 - 2i + 2i - 4i²Again, i² = -1, so -4i² = 4.Combine like terms: 1 + 4 = 5, and -2i + 2i = 0.So, denominator is 5.Therefore, (overline{z} = frac{10 - 5i}{5} = 2 - i).Alright, so (overline{z} = 2 - i). Since the conjugate of z is 2 - i, that means z must be 2 + i. Because if (overline{z} = a - bi), then z = a + bi. So, z = 2 + i.Now, the second part is to find (frac{z}{overline{z}}). Let's plug in the values we have:[frac{z}{overline{z}} = frac{2 + i}{2 - i}]Again, this is a complex division, so I should multiply numerator and denominator by the conjugate of the denominator to simplify. The conjugate of (2 - i) is (2 + i). So:[frac{2 + i}{2 - i} times frac{2 + i}{2 + i} = frac{(2 + i)(2 + i)}{(2 - i)(2 + i)}]Let me compute the numerator and denominator separately.First, the numerator: (2 + i)(2 + i)2*2 = 42*i = 2ii*2 = 2ii*i = i² = -1Adding them up: 4 + 2i + 2i - 1 = 3 + 4iDenominator: (2 - i)(2 + i)2*2 = 42*i = 2i(-i)*2 = -2i(-i)*i = -i² = 1 (since i² = -1)Adding them up: 4 + 2i - 2i + 1 = 5So, putting it together:[frac{3 + 4i}{5} = frac{3}{5} + frac{4}{5}i]Therefore, (frac{z}{overline{z}} = frac{3}{5} + frac{4}{5}i).Let me just double-check my calculations to make sure I didn't make any mistakes.Starting with (overline{z}):(4 + 3i)/(1 + 2i) multiplied by (1 - 2i)/(1 - 2i) gives (10 - 5i)/5 = 2 - i. That seems correct.Then, z is 2 + i, which makes sense.For the ratio z/(overline{z}):(2 + i)/(2 - i) multiplied by (2 + i)/(2 + i) gives (3 + 4i)/5, which is 3/5 + 4/5i. That looks right.I think that's all. I don't see any errors in my steps.**Final Answer**The complex number ( z ) is (boxed{2 + i}) and the ratio (dfrac{z}{overline{z}}) is (boxed{dfrac{3}{5} + dfrac{4}{5}i}).

answer：To express 1.14 billion in scientific notation, we start by recognizing that "billion" corresponds to (10^9). 1.14 billion can be written as (1.14 times 10^9).Thus, the correct answer is ( boxed{text{C: } 1.14 times 10^9} ).

answer：First, I need to find the mean of the numbers 8, 15, and 21. To do this, I will add these numbers together and then divide by 3.Next, I know that the mean of 16, z, and y is the same as the mean I just calculated. Since z is given as 24, I can substitute that value into the equation.Now, I will set up the equation for the mean of 16, 24, and y and set it equal to the mean of the first set of numbers.To solve for y, I will multiply both sides of the equation by 3 to eliminate the denominator.After simplifying, I will isolate y by subtracting the sum of 16 and 24 from both sides of the equation.Finally, I will calculate the value of y.

answer：Okay, so I have this problem where I need to find the equation of the nine-point circle, also known as the Feuerbach circle, of a triangle with vertices at (a, 0), (b, 0), and (0, c). Hmm, I remember that the nine-point circle passes through nine significant points of a triangle, including the midpoints of the sides, the feet of the altitudes, and the midpoints between the orthocenter and each vertex. But I'm not entirely sure about all the details, so I need to think carefully.First, maybe I should recall the general equation of a circle. It's usually written as ( x^2 + y^2 + Dx + Ey + F = 0 ), where D, E, and F are constants. Since the nine-point circle passes through the midpoints of the sides, I can find those midpoints and then plug them into the circle equation to solve for D, E, and F.Let me find the midpoints of the sides. The triangle has three sides: between (a, 0) and (b, 0), between (b, 0) and (0, c), and between (0, c) and (a, 0). 1. Midpoint between (a, 0) and (b, 0): The midpoint formula is ( left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2} right) ). So, this midpoint is ( left( frac{a + b}{2}, 0 right) ).2. Midpoint between (b, 0) and (0, c): Applying the midpoint formula, we get ( left( frac{b + 0}{2}, frac{0 + c}{2} right) = left( frac{b}{2}, frac{c}{2} right) ).3. Midpoint between (0, c) and (a, 0): Similarly, this is ( left( frac{0 + a}{2}, frac{c + 0}{2} right) = left( frac{a}{2}, frac{c}{2} right) ).Okay, so now I have three midpoints: ( left( frac{a + b}{2}, 0 right) ), ( left( frac{b}{2}, frac{c}{2} right) ), and ( left( frac{a}{2}, frac{c}{2} right) ). These three points should lie on the nine-point circle, so I can substitute each into the general circle equation to get three equations.Let me write down the general equation again: ( x^2 + y^2 + Dx + Ey + F = 0 ).1. Substituting ( left( frac{a + b}{2}, 0 right) ): ( left( frac{a + b}{2} right)^2 + 0^2 + D left( frac{a + b}{2} right) + E(0) + F = 0 ) Simplifying: ( frac{(a + b)^2}{4} + frac{D(a + b)}{2} + F = 0 ) --- Equation (1)2. Substituting ( left( frac{b}{2}, frac{c}{2} right) ): ( left( frac{b}{2} right)^2 + left( frac{c}{2} right)^2 + D left( frac{b}{2} right) + E left( frac{c}{2} right) + F = 0 ) Simplifying: ( frac{b^2}{4} + frac{c^2}{4} + frac{D b}{2} + frac{E c}{2} + F = 0 ) --- Equation (2)3. Substituting ( left( frac{a}{2}, frac{c}{2} right) ): ( left( frac{a}{2} right)^2 + left( frac{c}{2} right)^2 + D left( frac{a}{2} right) + E left( frac{c}{2} right) + F = 0 ) Simplifying: ( frac{a^2}{4} + frac{c^2}{4} + frac{D a}{2} + frac{E c}{2} + F = 0 ) --- Equation (3)Now, I have three equations: (1), (2), and (3). I need to solve this system for D, E, and F.Let me subtract Equation (3) from Equation (2) to eliminate some variables.Equation (2) - Equation (3):( left( frac{b^2}{4} + frac{c^2}{4} + frac{D b}{2} + frac{E c}{2} + F right) - left( frac{a^2}{4} + frac{c^2}{4} + frac{D a}{2} + frac{E c}{2} + F right) = 0 )Simplifying:( frac{b^2 - a^2}{4} + frac{D(b - a)}{2} = 0 )Factor the numerator:( frac{(b - a)(b + a)}{4} + frac{D(b - a)}{2} = 0 )Factor out (b - a):( (b - a) left( frac{b + a}{4} + frac{D}{2} right) = 0 )Assuming ( b neq a ) (since otherwise, the triangle would be degenerate), we can divide both sides by (b - a):( frac{b + a}{4} + frac{D}{2} = 0 )Solving for D:( frac{D}{2} = -frac{a + b}{4} )Multiply both sides by 2:( D = -frac{a + b}{2} )Okay, so D is found. Now, let's plug D back into Equation (1) to find F.Equation (1):( frac{(a + b)^2}{4} + frac{D(a + b)}{2} + F = 0 )Substitute D:( frac{(a + b)^2}{4} + frac{ -frac{a + b}{2} (a + b) }{2} + F = 0 )Simplify the second term:( frac{(a + b)^2}{4} - frac{(a + b)^2}{4} + F = 0 )So, the first two terms cancel out:( 0 + F = 0 )Thus, F = 0.Alright, now we have D and F. Let's plug D and F into Equation (2) to find E.Equation (2):( frac{b^2}{4} + frac{c^2}{4} + frac{D b}{2} + frac{E c}{2} + F = 0 )Substitute D = - (a + b)/2 and F = 0:( frac{b^2}{4} + frac{c^2}{4} + frac{ -frac{a + b}{2} cdot b }{2} + frac{E c}{2} = 0 )Simplify term by term:First term: ( frac{b^2}{4} )Second term: ( frac{c^2}{4} )Third term: ( frac{ - (a + b) b }{4} = frac{ -ab - b^2 }{4} )Fourth term: ( frac{E c}{2} )Combine all terms:( frac{b^2}{4} + frac{c^2}{4} - frac{ab + b^2}{4} + frac{E c}{2} = 0 )Simplify the b^2 terms:( frac{b^2 - b^2}{4} = 0 )So, we have:( frac{c^2}{4} - frac{ab}{4} + frac{E c}{2} = 0 )Multiply the entire equation by 4 to eliminate denominators:( c^2 - ab + 2 E c = 0 )Solve for E:( 2 E c = ab - c^2 )( E = frac{ab - c^2}{2 c} )So, now we have D, E, and F:- D = - (a + b)/2- E = (ab - c^2)/(2 c)- F = 0Therefore, the equation of the nine-point circle is:( x^2 + y^2 + D x + E y + F = 0 )Substituting the values:( x^2 + y^2 - frac{a + b}{2} x + frac{ab - c^2}{2 c} y = 0 )Alternatively, we can write this as:( 2 c x^2 + 2 c y^2 - (a + b) c x + (ab - c^2) y = 0 )by multiplying both sides by 2 c to eliminate denominators.Let me double-check my steps to make sure I didn't make any mistakes. I found the midpoints correctly, substituted them into the general circle equation, solved the system of equations, and found D, E, F. The algebra seems correct, and each step follows logically. I think this should be the correct equation for the nine-point circle.**Final Answer**The equation of the nine-point circle is boxed{2cleft(x^{2} + y^{2}right) - (a+b)cx + (ab - c^{2})y = 0}.