answer：Okay, so I have this problem here about a complex number z. It says that |z - 3 - 4i| = 1, and I need to find the maximum value of |z|. The options are A: 3, B: 4, C: 5, D: 6. Hmm, let me think about how to approach this.First, I remember that complex numbers can be represented as points in the plane, where z = x + yi corresponds to the point (x, y). The modulus |z| is the distance from the point z to the origin (0, 0). So, |z| is like the length of the vector from the origin to the point z.Now, the equation |z - 3 - 4i| = 1. This looks like the equation of a circle in the complex plane. Specifically, it's the set of all points z that are at a distance of 1 from the point 3 + 4i. So, if I think of 3 + 4i as a point (3, 4) in the plane, then |z - 3 - 4i| = 1 is a circle centered at (3, 4) with a radius of 1.So, z lies somewhere on this circle. I need to find the maximum value of |z|, which is the distance from the origin to the point z on this circle. So, essentially, I need to find the farthest point from the origin on this circle.To visualize this, I can imagine the circle centered at (3, 4) with radius 1. The origin is at (0, 0). The distance from the origin to the center of the circle (3, 4) can be found using the distance formula. Let me calculate that.The distance between (0, 0) and (3, 4) is sqrt[(3 - 0)^2 + (4 - 0)^2] = sqrt[9 + 16] = sqrt[25] = 5. So, the origin is 5 units away from the center of the circle.Since the circle has a radius of 1, the farthest point from the origin on this circle would be in the direction away from the origin. That is, if I go from the origin towards the center of the circle, and then extend that line by the radius of the circle, I'll reach the farthest point.So, the maximum distance from the origin to a point on the circle is the distance from the origin to the center plus the radius of the circle. That would be 5 + 1 = 6.Wait, let me make sure I'm not making a mistake here. Is it always just the distance from the origin to the center plus the radius? I think so, because the farthest point would be colinear with the origin and the center, lying on the circumference of the circle.Let me think of another way to approach this. Maybe using coordinates. Let me denote z = x + yi, so |z - 3 - 4i| = 1 translates to sqrt[(x - 3)^2 + (y - 4)^2] = 1. Squaring both sides, we get (x - 3)^2 + (y - 4)^2 = 1.We need to maximize |z|, which is sqrt(x^2 + y^2). To maximize sqrt(x^2 + y^2), we can instead maximize x^2 + y^2, since sqrt is an increasing function.So, the problem reduces to finding the maximum of x^2 + y^2 subject to (x - 3)^2 + (y - 4)^2 = 1.This is a constrained optimization problem. I can use the method of Lagrange multipliers, but maybe there's a simpler way.Alternatively, I can parametrize the circle. Since (x - 3)^2 + (y - 4)^2 = 1, I can write x = 3 + cos(theta), y = 4 + sin(theta), where theta is the parameter varying from 0 to 2π.Then, |z|^2 = x^2 + y^2 = (3 + cos(theta))^2 + (4 + sin(theta))^2.Let me expand that:(3 + cos(theta))^2 = 9 + 6 cos(theta) + cos^2(theta)(4 + sin(theta))^2 = 16 + 8 sin(theta) + sin^2(theta)Adding them together:9 + 6 cos(theta) + cos^2(theta) + 16 + 8 sin(theta) + sin^2(theta) = 25 + 6 cos(theta) + 8 sin(theta) + (cos^2(theta) + sin^2(theta)).But cos^2(theta) + sin^2(theta) = 1, so this simplifies to 25 + 6 cos(theta) + 8 sin(theta) + 1 = 26 + 6 cos(theta) + 8 sin(theta).So, |z|^2 = 26 + 6 cos(theta) + 8 sin(theta). To find the maximum of |z|, we need to maximize |z|^2, which is 26 + 6 cos(theta) + 8 sin(theta).The maximum of 6 cos(theta) + 8 sin(theta) can be found using the amplitude formula. The maximum of A cos(theta) + B sin(theta) is sqrt(A^2 + B^2). So, here, A = 6, B = 8, so the maximum is sqrt(36 + 64) = sqrt(100) = 10.Therefore, the maximum of |z|^2 is 26 + 10 = 36. Hence, the maximum of |z| is sqrt(36) = 6.So, that confirms my earlier thought. The maximum value of |z| is 6, which is option D.Wait, just to make sure, let me think if there's another approach. Maybe using vectors or geometry.Imagine the origin, the center of the circle at (3,4), and the circle itself. The maximum |z| is the distance from the origin to the farthest point on the circle. Since the circle is centered at (3,4) with radius 1, the farthest point from the origin would be along the line connecting the origin to the center of the circle, extended by the radius.So, the distance from the origin to the center is 5, as I calculated before. Adding the radius 1, we get 6. So, that's consistent with the previous methods.Alternatively, if I think of the triangle formed by the origin, the center of the circle, and a point z on the circle. The sides of the triangle are 5 (from origin to center), 1 (radius), and |z| (from origin to z). By the triangle inequality, |z| <= 5 + 1 = 6, and equality occurs when z is colinear with the origin and the center, on the same side as the center.So, all these methods lead me to the conclusion that the maximum value of |z| is 6. Therefore, the correct answer is D: 6.**Final Answer**boxed{D}

answer：Okay, so I have this problem where I need to find the intersection of two sets, M and N. Let me see... Set M is defined as all real numbers x such that 0 is less than or equal to x and x is less than 3. So, in interval notation, that would be [0, 3). Got that down.Now, set N is defined by the inequality x squared minus 3x minus 4 is less than zero. Hmm, quadratic inequality. I remember that to solve quadratic inequalities, I need to factor the quadratic if possible, find the roots, and then determine the intervals where the inequality holds true.Let me try factoring x² - 3x - 4. I'm looking for two numbers that multiply to -4 and add up to -3. Let's see... 1 and -4? Yes, because 1 times -4 is -4 and 1 plus (-4) is -3. So, the factored form should be (x + 1)(x - 4) < 0. Okay, that seems right.Now, to solve (x + 1)(x - 4) < 0, I need to find where this product is negative. I know that the product of two numbers is negative when one is positive and the other is negative. So, I can use a sign chart or test intervals around the critical points, which are x = -1 and x = 4.Let me list the critical points: x = -1 and x = 4. These divide the real number line into three intervals:1. x < -12. -1 < x < 43. x > 4I'll test each interval to see where the product (x + 1)(x - 4) is negative.First interval: x < -1. Let's pick x = -2. Plugging in, (-2 + 1)(-2 - 4) = (-1)(-6) = 6, which is positive. So, this interval doesn't satisfy the inequality.Second interval: -1 < x < 4. Let's choose x = 0. (0 + 1)(0 - 4) = (1)(-4) = -4, which is negative. So, this interval satisfies the inequality.Third interval: x > 4. Let's take x = 5. (5 + 1)(5 - 4) = (6)(1) = 6, which is positive. So, this interval doesn't satisfy the inequality either.Therefore, the solution to the inequality x² - 3x - 4 < 0 is -1 < x < 4. So, set N is (-1, 4).Now, I need to find the intersection of sets M and N. Set M is [0, 3) and set N is (-1, 4). The intersection of these two sets will be the overlap where both conditions are satisfied.Visualizing the intervals on a number line:- Set M starts at 0 (inclusive) and goes up to, but not including, 3.- Set N starts just above -1 and goes up to, but not including, 4.So, the overlap between M and N would start at 0 (since that's where M starts) and go up to 3 (since that's where M ends). Therefore, the intersection should be [0, 3).Let me double-check to make sure I didn't make a mistake. Set N is (-1, 4), which includes all numbers from just above -1 to just below 4. Set M is [0, 3), which includes 0 up to but not including 3. So, the overlap is indeed from 0 to 3, including 0 but not 3. That seems correct.I think I got it. The intersection of M and N is [0, 3), which corresponds to option C.

answer：First, I need to determine the number of apples in the bowl. Let's denote the number of apples as ( A ).The bowl initially contains 23 oranges. If 13 oranges are removed, the number of oranges left will be ( 23 - 13 = 10 ) oranges.After removing the oranges, the total number of pieces of fruit in the bowl will be the sum of apples and the remaining oranges, which is ( A + 10 ).According to the problem, the apples should make up 50% of the total pieces of fruit after the oranges are removed. This means:[A = 0.5 times (A + 10)]To solve for ( A ), I'll first multiply both sides of the equation by 2 to eliminate the decimal:[2A = A + 10]Next, I'll subtract ( A ) from both sides to isolate ( A ):[2A - A = 10][A = 10]Therefore, there are 10 apples in the bowl.

answer：Alright, so I have this problem here where I need to figure out which equation correctly represents the y-values for the given x-values in a table. The table is:[begin{tabular}[t]{|c|c|c|c|c|c|}hline x & 0 & 1 & 2 & 3 & 4hline y & 200 & 140 & 80 & 20 & 0hline end{tabular}]And the options are:[textbf{(A)} y=200-15xqquadtextbf{(B)} y=200-20x+5x^{2}qquadtextbf{(C)} y=200-30x+10x^{2}qquadtextbf{(D)} y=150-50xqquadtextbf{(E)} text{None of these}]Okay, so I need to determine which equation correctly models the relationship between x and y based on the table provided. Let's break this down step by step.First, I'll look at the simplest option, which is option A: ( y = 200 - 15x ). This is a linear equation, meaning it has a constant rate of change. Let's test this equation with the given x-values to see if it matches the y-values in the table.Starting with x = 0:[y = 200 - 15(0) = 200]That's correct because the table shows y = 200 when x = 0.Next, x = 1:[y = 200 - 15(1) = 185]But the table shows y = 140 when x = 1. Hmm, that's not matching. So, option A might not be the correct equation.Wait, maybe I made a mistake. Let me double-check:[y = 200 - 15(1) = 185]No, that's correct. So, option A doesn't match the y-value for x = 1.Okay, moving on to option B: ( y = 200 - 20x + 5x^2 ). This is a quadratic equation, so it has a changing rate of change. Let's test this with the same x-values.Starting with x = 0:[y = 200 - 20(0) + 5(0)^2 = 200]That's correct.Next, x = 1:[y = 200 - 20(1) + 5(1)^2 = 200 - 20 + 5 = 185]Again, the table shows y = 140, so this doesn't match either.Hmm, maybe I need to try another approach. Perhaps I should calculate the differences between consecutive y-values to see if there's a pattern.Looking at the y-values: 200, 140, 80, 20, 0.Calculating the first differences (the change in y for each increment in x):- From x=0 to x=1: 140 - 200 = -60- From x=1 to x=2: 80 - 140 = -60- From x=2 to x=3: 20 - 80 = -60- From x=3 to x=4: 0 - 20 = -20Wait, the first differences are not constant; they are -60, -60, -60, -20. That doesn't make sense because if it were a linear relationship, the first differences should be constant. Since they're not, maybe it's a quadratic relationship.For a quadratic relationship, the second differences should be constant. Let's calculate the second differences:- First differences: -60, -60, -60, -20- Second differences: (-60 - (-60)) = 0, (-60 - (-60)) = 0, (-20 - (-60)) = 40Hmm, the second differences are 0, 0, 40, which are not constant either. That's confusing. Maybe it's not a quadratic relationship then.Wait, perhaps I made a mistake in calculating the differences. Let me try again.First differences:- From x=0 to x=1: 140 - 200 = -60- From x=1 to x=2: 80 - 140 = -60- From x=2 to x=3: 20 - 80 = -60- From x=3 to x=4: 0 - 20 = -20Second differences:- Between first and second difference: -60 - (-60) = 0- Between second and third difference: -60 - (-60) = 0- Between third and fourth difference: -20 - (-60) = 40Yes, that's correct. The second differences are 0, 0, 40, which are not constant. So, it's not a quadratic relationship either.Maybe I need to consider a different type of function. Let's look at the options again.Option C: ( y = 200 - 30x + 10x^2 )Let's test this equation with the x-values.Starting with x = 0:[y = 200 - 30(0) + 10(0)^2 = 200]Correct.x = 1:[y = 200 - 30(1) + 10(1)^2 = 200 - 30 + 10 = 180]But the table shows y = 140. Not matching.x = 2:[y = 200 - 30(2) + 10(2)^2 = 200 - 60 + 40 = 180]Table shows y = 80. Not matching.Hmm, not matching either.Option D: ( y = 150 - 50x )Testing this:x = 0:[y = 150 - 50(0) = 150]Table shows y = 200. Not matching.So, none of the options A to D seem to match the given y-values correctly. Therefore, the correct answer might be option E: None of these.But wait, let me double-check my calculations to make sure I didn't make any mistakes.For option A:- x=0: 200- x=1: 185- x=2: 170- x=3: 155- x=4: 140None of these match the table except x=0.Option B:- x=0: 200- x=1: 185- x=2: 180- x=3: 185- x=4: 200Again, only x=0 matches.Option C:- x=0: 200- x=1: 180- x=2: 180- x=3: 200- x=4: 240None match except x=0.Option D:- x=0: 150- x=1: 100- x=2: 50- x=3: 0- x=4: -50None match.So, indeed, none of the options A to D correctly represent the y-values for the given x-values. Therefore, the correct answer is E: None of these.

answer：Okay, so I have this problem about a point A on a parabola. The parabola is given by the equation ( y^2 = 2px ) where ( p > 0 ). I need to find the distance from the origin O to the point A, which is denoted as ( |OA| ). First, let me recall some properties of a parabola. The standard form of a parabola that opens to the right is ( y^2 = 4ax ), where the focus is at ( (a, 0) ). Comparing this with the given equation ( y^2 = 2px ), it seems like ( 4a = 2p ), so ( a = frac{p}{2} ). That means the focus of this parabola is at ( left( frac{p}{2}, 0 right) ).Now, the problem states that the distance from point A to the focus is 6. Let me denote the coordinates of point A as ( (x, y) ). The distance from A to the focus can be calculated using the distance formula:[sqrt{left( x - frac{p}{2} right)^2 + y^2} = 6]Also, it's given that the distance from point A to the y-axis is 3. The distance from a point ( (x, y) ) to the y-axis is simply the absolute value of its x-coordinate, so:[|x| = 3]Since the parabola ( y^2 = 2px ) opens to the right and ( p > 0 ), the x-coordinate of any point on the parabola must be non-negative. Therefore, ( x = 3 ).Now, I can substitute ( x = 3 ) into the equation of the parabola to find y:[y^2 = 2p times 3 = 6p]So, ( y = pm sqrt{6p} ).Next, I'll use the distance from A to the focus. We have:[sqrt{left( 3 - frac{p}{2} right)^2 + (sqrt{6p})^2} = 6]Let me square both sides to eliminate the square root:[left( 3 - frac{p}{2} right)^2 + 6p = 36]Expanding the first term:[left( 3 - frac{p}{2} right)^2 = 9 - 3p + frac{p^2}{4}]So, substituting back:[9 - 3p + frac{p^2}{4} + 6p = 36]Combine like terms:[9 + 3p + frac{p^2}{4} = 36]Subtract 36 from both sides:[frac{p^2}{4} + 3p + 9 - 36 = 0]Simplify:[frac{p^2}{4} + 3p - 27 = 0]Multiply every term by 4 to eliminate the fraction:[p^2 + 12p - 108 = 0]Now, I have a quadratic equation in terms of p:[p^2 + 12p - 108 = 0]I can solve this using the quadratic formula:[p = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Where ( a = 1 ), ( b = 12 ), and ( c = -108 ). Plugging these in:[p = frac{-12 pm sqrt{144 + 432}}{2}][p = frac{-12 pm sqrt{576}}{2}][p = frac{-12 pm 24}{2}]So, we have two solutions:1. ( p = frac{-12 + 24}{2} = frac{12}{2} = 6 )2. ( p = frac{-12 - 24}{2} = frac{-36}{2} = -18 )But since ( p > 0 ), we discard the negative solution. Therefore, ( p = 6 ).Now, knowing that ( p = 6 ), we can find the coordinates of point A. We already know ( x = 3 ). Let's find y:[y^2 = 6p = 6 times 6 = 36][y = pm 6]So, point A can be either ( (3, 6) ) or ( (3, -6) ). Since distance is a positive quantity, both points will give the same distance from the origin.Now, let's compute ( |OA| ), the distance from the origin ( (0, 0) ) to point A ( (3, 6) ) or ( (3, -6) ):Using the distance formula:[|OA| = sqrt{(3 - 0)^2 + (6 - 0)^2} = sqrt{9 + 36} = sqrt{45} = 3sqrt{5}]Alternatively, for ( (3, -6) ):[|OA| = sqrt{(3 - 0)^2 + (-6 - 0)^2} = sqrt{9 + 36} = sqrt{45} = 3sqrt{5}]So, in both cases, the distance is ( 3sqrt{5} ).Let me double-check my steps to ensure I didn't make any mistakes.1. I found the focus correctly at ( left( frac{p}{2}, 0 right) ).2. I used the distance formula correctly for the distance from A to the focus.3. I correctly noted that the distance from A to the y-axis is 3, so ( x = 3 ).4. Substituted ( x = 3 ) into the parabola equation to find ( y^2 = 6p ).5. Plugged ( x = 3 ) and ( y^2 = 6p ) into the distance formula for the focus, squared both sides, expanded, and solved the quadratic equation correctly.6. Discarded the negative solution for p since ( p > 0 ).7. Calculated y correctly as ( pm 6 ).8. Applied the distance formula from the origin correctly, resulting in ( 3sqrt{5} ).Everything seems to check out. I think the answer is ( 3sqrt{5} ), which corresponds to option C.**Final Answer**boxed{C}

answer：First, I need to determine the total amount of clay Anne uses to make her plates. Since Anne makes 30 rectangular plates and each plate has the same thickness, the total volume of clay she uses is proportional to the total area of her plates.Next, I'll assume that each of Anne's rectangular plates has a specific area. For simplicity, let's say each rectangular plate has dimensions of 4 units by 5 units, resulting in an area of 20 square units per plate. Therefore, the total area of all 30 plates is 30 multiplied by 20, which equals 600 square units.Since all friends use the same amount of clay, Bob also uses 600 square units of clay for his circular plates. If Bob makes 15 circular plates, the area of one circular plate is 600 divided by 15, which equals 40 square units.Anne sells each of her plates for 50 cents, so her total earnings from 30 plates are 30 multiplied by 50 cents, which equals 1500 cents. To earn the same amount, Bob needs to set the price of his 15 plates such that the total earnings are also 1500 cents. Therefore, the price per plate for Bob is 1500 cents divided by 15 plates, which equals 100 cents per plate.