💡Okay, so I need to find the minimum possible value for y in the equation y equals x squared plus 16x plus 20. Hmm, I remember that quadratic equations graph as parabolas, and since the coefficient of x squared is positive, this parabola opens upwards. That means the vertex of the parabola will be its lowest point, so the y-value at the vertex is the minimum value of y.Alright, to find the vertex, I think there are a couple of methods. One is completing the square, which I've heard about but I'm not super confident about. The other is using the formula for the vertex. I think the formula is something like x equals negative b over 2a for a quadratic equation in standard form, which is ax squared plus bx plus c.Let me try that formula first because it seems straightforward. In my equation, a is 1 because the coefficient of x squared is 1, b is 16, and c is 20. So plugging into the formula, x equals negative 16 divided by 2 times 1. That simplifies to x equals negative 16 over 2, which is negative 8. Okay, so the x-coordinate of the vertex is negative 8.Now, to find the corresponding y-value, I need to plug x equals negative 8 back into the original equation. So y equals (-8) squared plus 16 times (-8) plus 20. Let's calculate that step by step.First, (-8) squared is 64. Then, 16 times (-8) is negative 128. So now, y equals 64 minus 128 plus 20. Let's do the math: 64 minus 128 is negative 64, and negative 64 plus 20 is negative 44. So the y-value at the vertex is negative 44.Wait, that seems pretty straightforward. But just to make sure I didn't make any mistakes, maybe I should try completing the square as another method to confirm the result.Starting with the original equation: y equals x squared plus 16x plus 20. To complete the square, I need to make the quadratic expression into a perfect square trinomial. The coefficient of x squared is 1, so that's good. I take the coefficient of x, which is 16, divide it by 2 to get 8, and then square it to get 64. So I add and subtract 64 inside the equation.That gives me y equals (x squared plus 16x plus 64) minus 64 plus 20. Simplifying that, the expression inside the parentheses is a perfect square: (x plus 8) squared. So now, y equals (x plus 8) squared minus 44.Since (x plus 8) squared is always non-negative, the smallest value it can take is 0. That happens when x plus 8 equals 0, which is when x equals negative 8. Plugging that back in, y equals 0 minus 44, which is negative 44. So that confirms my earlier result.Hmm, both methods give me the same answer, so I feel more confident that negative 44 is indeed the minimum value of y. I guess it's always good to check using different methods to avoid mistakes.Just to visualize it, if I imagine the graph of this quadratic, it's a parabola opening upwards with its vertex at (-8, -44). So the lowest point on this graph is at y equals negative 44, which is the minimum value. If I were to plot some points around x equals negative 8, I should see that y increases as I move left or right from that point.For example, if x is negative 7, then y equals (-7)^2 plus 16*(-7) plus 20. That's 49 minus 112 plus 20, which is 49 minus 112 is negative 63, plus 20 is negative 43. So y is negative 43 when x is negative 7, which is higher than negative 44.Similarly, if x is negative 9, y equals (-9)^2 plus 16*(-9) plus 20. That's 81 minus 144 plus 20, which is 81 minus 144 is negative 63, plus 20 is negative 43 again. So y is also negative 43 when x is negative 9, which is again higher than negative 44.This symmetry makes sense because the parabola is symmetric around the vertex. So the points equidistant from the vertex on either side have the same y-value, which is higher than the vertex's y-value.I think I've covered all the bases here. I used two different methods to find the minimum value, checked my work by plugging in nearby x-values, and even thought about the graph to visualize why the minimum occurs at that point. I don't see any mistakes in my calculations, so I'm pretty sure that the minimum possible value for y is negative 44.

💡Okay, so I have this geometry problem here about an obtuse triangle ABC where angle C is greater than 90 degrees. They tell me that AM equals MB, which means M is the midpoint of AB. Then, MD is perpendicular to BC, and EC is also perpendicular to BC. D is on BC, E is on AC, and M is the midpoint of AB. The area of triangle ABC is 30, and I need to find the area of triangle BED. The options are A) 10, B) 12, C) 15, D) 18, E) 20.Alright, let me try to visualize this triangle. Since angle C is obtuse, triangle ABC is an obtuse triangle with the obtuse angle at C. So, sides AC and BC are the legs, and AB is the base. M is the midpoint of AB, so AM equals MB. Then, from M, we drop a perpendicular to BC, which is MD. Also, from E on AC, we drop a perpendicular to BC, which is EC.First, maybe I should draw a rough sketch of this triangle. Let me imagine point C at the top, making an obtuse angle, with points A and B at the base. M is the midpoint of AB, so it's right in the middle of the base. Then, from M, I draw MD perpendicular to BC, so D is somewhere on BC. From E on AC, I draw EC perpendicular to BC, so E is somewhere on AC.Given that the area of triangle ABC is 30, I can use that to find some relationships. The area of a triangle is (1/2)*base*height. If I take BC as the base, then the height would be from A perpendicular to BC. But in this case, EC is perpendicular to BC, so EC is another height from E to BC.Wait, maybe I can express the area of ABC in terms of BC and the height from A. Let me denote BC as 'a', AB as 'c', and AC as 'b'. The area is 30, so (1/2)*a*height from A = 30. But I don't know the height from A yet.Alternatively, since EC is perpendicular to BC, EC is the height from E to BC. Maybe I can relate EC to the height from A. Since E is on AC, perhaps EC is a fraction of the height from A.Also, MD is perpendicular to BC, so MD is the height from M to BC. Since M is the midpoint of AB, maybe MD is half the height from A? Hmm, not sure if that's correct because the height from M might not necessarily be half of the height from A. It depends on the triangle's shape.Let me think about coordinates. Maybe assigning coordinates to the triangle can help me compute the areas. Let me place point C at (0,0) since it's the vertex with the obtuse angle. Then, since angle C is greater than 90 degrees, points A and B will be in such a way that the triangle is obtuse at C.Wait, actually, if I place C at (0,0), and BC along the x-axis, then point B can be at (a,0), and point A somewhere in the plane. Since angle C is obtuse, point A should be such that the angle at C is greater than 90 degrees. So, if I place point C at (0,0), point B at (b,0), and point A at (d,e), then the vectors CA and CB should form an angle greater than 90 degrees.But maybe coordinates will complicate things. Let me try another approach.Since M is the midpoint of AB, and MD is perpendicular to BC, maybe I can find the length of MD in terms of the height of the triangle. Let me denote the height from A to BC as h. Then, the area of ABC is (1/2)*BC*h = 30, so BC*h = 60.Now, since M is the midpoint of AB, the height from M to BC, which is MD, should be half of the height from A to BC. Is that correct? Wait, no, that might not be the case because the height depends on the position of M.Alternatively, maybe I can use similar triangles. Since MD is perpendicular to BC and EC is also perpendicular to BC, both MD and EC are heights onto BC. Maybe triangles MBD and EBC are similar? Let me see.Wait, triangle MBD: M is the midpoint of AB, MD is perpendicular to BC. Triangle EBC: E is on AC, EC is perpendicular to BC. Maybe there's a similarity here.Alternatively, perhaps I can express BD and BE in terms of the sides of the triangle and then compute the area of BED.Wait, let me think step by step.1. Let me denote BC as the base, length 'a'. The height from A to BC is 'h', so area of ABC is (1/2)*a*h = 30, so a*h = 60.2. M is the midpoint of AB, so AM = MB = c/2, where c is the length of AB.3. MD is perpendicular to BC, so MD is the height from M to BC. Similarly, EC is perpendicular to BC, so EC is the height from E to BC.4. Since E is on AC, and EC is perpendicular to BC, E is the foot of the perpendicular from C to AC? Wait, no, EC is perpendicular to BC, so E is the foot of the perpendicular from E to BC. Wait, that doesn't make sense. Wait, EC is perpendicular to BC, so EC is the height from E to BC, meaning E is on AC, and EC is perpendicular to BC.Wait, maybe I can express EC in terms of the coordinates.Alternatively, let me assign coordinates to the triangle to make it easier.Let me place point C at (0,0), point B at (a,0), and point A at (d,e). Since angle C is obtuse, the dot product of vectors CA and CB should be negative.Vectors CA = (d,e) and CB = (a,0). The dot product is d*a + e*0 = d*a. For angle C to be obtuse, the dot product should be negative, so d*a < 0. Therefore, d and a have opposite signs. Since a is the length BC, which is positive, d must be negative. So, point A is at (d,e) where d is negative.Now, M is the midpoint of AB. Coordinates of A are (d,e), coordinates of B are (a,0). So, midpoint M has coordinates ((d+a)/2, e/2).Now, MD is perpendicular to BC. Since BC is along the x-axis from (0,0) to (a,0), the line BC is horizontal. Therefore, MD is a vertical line from M down to BC. So, the x-coordinate of D is the same as that of M, which is (d+a)/2, and y-coordinate is 0. So, point D is ((d+a)/2, 0).Similarly, EC is perpendicular to BC. Since BC is horizontal, EC is vertical. So, EC is a vertical line from E on AC down to BC. So, point E has the same x-coordinate as point C, which is 0, but wait, no. Wait, EC is perpendicular to BC, which is horizontal, so EC is vertical. Therefore, E must lie somewhere on AC such that the line from E to C is vertical. But point C is at (0,0), so if EC is vertical, then E must have the same x-coordinate as C, which is 0. But E is on AC, so E is at (0, k) for some k.Wait, that can't be because AC goes from (d,e) to (0,0). So, the line AC has the equation y = (e/d)x. So, if E is on AC and EC is vertical, then E must be at (0,0), which is point C. But that can't be because E is on AC, not coinciding with C. Wait, maybe I made a mistake.Wait, EC is perpendicular to BC, which is horizontal, so EC is vertical. Therefore, EC is a vertical line from E to BC. Since BC is along the x-axis, EC must be a vertical line segment from E to the x-axis. Therefore, E must have the same x-coordinate as C, which is 0, but E is on AC.Wait, AC goes from (d,e) to (0,0). So, the line AC is y = (e/d)x. So, if E is on AC and EC is vertical, then E must be at (0,0), which is point C. But that can't be because E is on AC, not coinciding with C. So, maybe my coordinate system is not the best.Alternatively, maybe I should place point C at (0,0), point B at (b,0), and point A at (0,a), making triangle ABC with right angle at C, but since it's obtuse, maybe point A is at (-a, b) or something.Wait, perhaps I'm overcomplicating. Let me try a different approach.Since EC is perpendicular to BC, and E is on AC, then EC is the altitude from E to BC. Similarly, MD is the altitude from M to BC.Given that, maybe I can express the lengths of EC and MD in terms of the sides of the triangle.Let me denote BC as 'a', AB as 'c', and AC as 'b'. The area of ABC is 30, so (1/2)*a*h = 30, where h is the height from A to BC. So, a*h = 60.Now, since M is the midpoint of AB, the coordinates of M would be the average of A and B. If I consider BC as the base, then the height from M to BC would be half the height from A to BC, right? Because M is halfway along AB.Wait, is that true? If I have a triangle and take the midpoint of the base, the height from the midpoint would be half the height from the opposite vertex. But in this case, M is the midpoint of AB, not the base BC. So, maybe the height from M to BC is not necessarily half of h.Hmm, maybe I need to use coordinate geometry after all.Let me place point C at (0,0), point B at (a,0), and point A at (0,b), making triangle ABC with right angle at C. But since it's obtuse at C, point A should be in such a way that angle C is greater than 90 degrees. So, maybe point A is at (-c, d), where c and d are positive.Wait, let me try this. Let me place point C at (0,0), point B at (c,0), and point A at (-d,e), where d and e are positive. Then, vectors CA = (-d,e) and CB = (c,0). The dot product is (-d)*c + e*0 = -dc. For angle C to be obtuse, the dot product should be negative, which it is since d and c are positive. So, that works.Now, the area of triangle ABC is 30. The area can be calculated using the determinant formula: (1/2)*|CA x CB| = 30. The cross product CA x CB is (-d)(0) - (e)(c) = -ec. So, the area is (1/2)*| -ec | = (1/2)*ec = 30. Therefore, ec = 60.Now, M is the midpoint of AB. Coordinates of A are (-d,e), coordinates of B are (c,0). So, midpoint M has coordinates ((-d + c)/2, (e + 0)/2) = ((c - d)/2, e/2).Now, MD is perpendicular to BC. Since BC is along the x-axis from (0,0) to (c,0), the line BC is horizontal. Therefore, MD is a vertical line from M down to BC. So, the x-coordinate of D is the same as that of M, which is (c - d)/2, and y-coordinate is 0. So, point D is ((c - d)/2, 0).Similarly, EC is perpendicular to BC. Since BC is horizontal, EC is vertical. So, EC is a vertical line from E on AC down to BC. Therefore, E must have the same x-coordinate as C, which is 0, but E is on AC.Wait, AC goes from (-d,e) to (0,0). The parametric equation of AC can be written as x = -d + t*d, y = e - t*e, where t ranges from 0 to 1.Wait, no, parametric equations for AC: starting at A (-d,e) and ending at C (0,0). So, the parametric equations are x = -d + t*d, y = e - t*e, where t goes from 0 to 1.But EC is vertical, meaning that the x-coordinate of E is the same as that of C, which is 0. So, setting x = 0 in the parametric equation: 0 = -d + t*d => t = 1. But when t = 1, y = 0, which is point C. So, E is point C, which can't be because E is on AC but not coinciding with C.Wait, that suggests that EC is the same as CC, which is just a point. That can't be right. Maybe my coordinate system is causing confusion.Alternatively, perhaps EC is not from E to C, but EC is perpendicular to BC, meaning that EC is a vertical line from E to BC. So, E is on AC, and EC is the vertical segment from E to BC. Therefore, E has the same x-coordinate as C, which is 0, but E is on AC.Wait, AC goes from (-d,e) to (0,0). So, if E is on AC and has x-coordinate 0, then E must be at (0,0), which is point C. That can't be because E is on AC but not coinciding with C. So, perhaps my initial assumption is wrong.Wait, maybe EC is not from E to C, but EC is a segment from E to BC, perpendicular to BC. So, E is on AC, and EC is the perpendicular from E to BC, landing at some point on BC, say point F. So, EC is perpendicular to BC, meaning that EC is vertical, and F is on BC.Wait, that makes more sense. So, E is on AC, and EC is the perpendicular from E to BC, landing at point F on BC. So, EC is the vertical segment from E to F on BC.But in that case, EC is not necessarily connected to point C. Wait, the problem says EC is perpendicular to BC, where E is on AC. So, EC is the segment from E to BC, perpendicular to BC. So, EC is the altitude from E to BC.Similarly, MD is the altitude from M to BC.So, in that case, EC is a vertical segment from E on AC down to BC, and MD is a vertical segment from M down to BC.Given that, maybe I can find the coordinates of E and D, then find the coordinates of B, E, D, and compute the area of triangle BED.Let me proceed step by step.1. Assign coordinates: - C at (0,0) - B at (c,0) - A at (-d,e), where d, e > 0 - M is the midpoint of AB: ((c - d)/2, e/2) - D is the foot of perpendicular from M to BC: ((c - d)/2, 0) - E is the foot of perpendicular from E to BC, so E is on AC, and EC is vertical.Wait, EC is vertical, so E has the same x-coordinate as C, which is 0. But E is on AC, which goes from (-d,e) to (0,0). So, the parametric equation of AC is x = -d + t*d, y = e - t*e, where t ∈ [0,1].If E is on AC and has x-coordinate 0, then:0 = -d + t*d => t = 1But when t = 1, y = 0, which is point C. So, E is point C, which can't be because E is on AC but not coinciding with C.Wait, this suggests that EC is not from E to C, but from E to BC, meaning that E is on AC, and EC is the perpendicular from E to BC, landing at some point F on BC. So, EC is the segment from E to F, where F is on BC, and EC is perpendicular to BC.In that case, EC is vertical, so F has the same x-coordinate as E. Since BC is along the x-axis, F is (x,0) where x is the x-coordinate of E.So, E is on AC, and EC is the vertical segment from E to F on BC. Therefore, E has coordinates (x,y), and F is (x,0). Since E is on AC, which goes from (-d,e) to (0,0), the parametric equation of AC is x = -d + t*d, y = e - t*e, t ∈ [0,1].So, E is (x,y) = (-d + t*d, e - t*e). Then, F is (x,0) = (-d + t*d, 0).But EC is the segment from E to F, which is vertical, so EC has length y = e - t*e.Now, I need to find t such that E is on AC and EC is perpendicular to BC. But since EC is vertical, it's already perpendicular to BC, which is horizontal. So, any point E on AC will have EC as vertical if E is projected down to BC. Wait, no, only if E is directly above F on BC.Wait, I think I'm overcomplicating. Let me think differently.Since EC is perpendicular to BC, and BC is horizontal, EC must be vertical. Therefore, E must lie directly above some point on BC. So, E is on AC, and the x-coordinate of E is the same as the x-coordinate of the foot of the perpendicular from E to BC.Therefore, E is on AC, and the x-coordinate of E is equal to the x-coordinate of F, where F is the foot of the perpendicular from E to BC.So, E is (x,y) on AC, and F is (x,0) on BC.Since E is on AC, which goes from (-d,e) to (0,0), the parametric equation is x = -d + t*d, y = e - t*e, t ∈ [0,1].So, E is (-d + t*d, e - t*e). Then, F is (-d + t*d, 0).But EC is the segment from E to F, which is vertical, so EC has length y = e - t*e.Now, I need to find t such that E is on AC and EC is perpendicular to BC. But since EC is vertical, it's already perpendicular to BC, which is horizontal. So, any E on AC will have EC as vertical if E is projected down to BC. Wait, no, only if E is directly above F on BC.Wait, I think I'm stuck here. Maybe I should use vectors or coordinate geometry to find the coordinates of E and D, then compute the area of BED.Given that, let's proceed.We have:- Point C: (0,0)- Point B: (c,0)- Point A: (-d,e)- Midpoint M: ((c - d)/2, e/2)- Point D: ((c - d)/2, 0) (since MD is vertical)- Point E: Let's find E on AC such that EC is vertical. Wait, as above, E must be at (x,y) where x is the same as F on BC, so x is the x-coordinate of E, and F is (x,0).But E is on AC, so E can be expressed as a point along AC. Let me parameterize AC.Parametric equations for AC:x = -d + t*d, y = e - t*e, where t ∈ [0,1].So, E is at (x,y) = (-d + t*d, e - t*e).Now, EC is the vertical segment from E to F on BC, so F is (x,0) = (-d + t*d, 0).But EC is perpendicular to BC, which is horizontal, so EC is vertical, which it is.Now, I need to find t such that E is on AC and EC is vertical. But as above, any E on AC will have EC vertical if E is projected down to BC. Wait, no, only if E is directly above F on BC.Wait, I think I'm going in circles. Maybe I can express the coordinates of E in terms of t and then find t.Alternatively, maybe I can find the ratio in which E divides AC.Let me denote the ratio AE:EC as k:1. Then, E divides AC in the ratio k:1. So, coordinates of E would be ((k*0 + 1*(-d))/(k + 1), (k*0 + 1*e)/(k + 1)) = (-d/(k + 1), e/(k + 1)).But EC is the vertical segment from E to BC, so EC has length y-coordinate of E, which is e/(k + 1).Now, I need to relate this to the area of triangle BED.Wait, maybe I can express the area of BED in terms of coordinates.Points:- B: (c,0)- E: (-d/(k + 1), e/(k + 1))- D: ((c - d)/2, 0)So, triangle BED has vertices at (c,0), (-d/(k + 1), e/(k + 1)), and ((c - d)/2, 0).To find the area of triangle BED, I can use the shoelace formula.First, let's list the coordinates:B: (c, 0)E: (-d/(k + 1), e/(k + 1))D: ((c - d)/2, 0)So, the area is (1/2)*| (c*(e/(k + 1) - 0) + (-d/(k + 1))*(0 - 0) + ((c - d)/2)*(0 - e/(k + 1)) ) |Simplify:= (1/2)*| c*(e/(k + 1)) + 0 + ((c - d)/2)*(-e/(k + 1)) |= (1/2)*| (c e)/(k + 1) - (c - d)e/(2(k + 1)) |= (1/2)*| (2c e - (c - d)e)/(2(k + 1)) |= (1/2)*| (2c e - c e + d e)/(2(k + 1)) |= (1/2)*| (c e + d e)/(2(k + 1)) |= (1/2)*( (c + d)e )/(2(k + 1))= (c + d)e/(4(k + 1))Now, we know that the area of ABC is 30, which is (1/2)*base*height = (1/2)*c*e = 30, so c*e = 60.Wait, no, in my coordinate system, the base BC is length c, and the height from A is e, because A is at (-d,e). So, the area is (1/2)*c*e = 30, so c*e = 60.So, (c + d)e/(4(k + 1)) = (c + d)*e/(4(k + 1)).But I need to express (c + d) in terms of known quantities.Wait, from the coordinates, point A is at (-d,e), point B is at (c,0), so AB has length sqrt( (c + d)^2 + e^2 ). But M is the midpoint of AB, which is ((c - d)/2, e/2).Wait, maybe I can find the relationship between d and c using the fact that angle C is obtuse.In triangle ABC, angle C is obtuse, so by the law of cosines:AB² > AC² + BC²So, (c + d)^2 + e^2 > (d^2 + e^2) + c^2Simplify:(c + d)^2 + e^2 > d^2 + e^2 + c^2Expanding (c + d)^2:c² + 2cd + d² + e² > d² + e² + c²Simplify:2cd > 0Since c and d are positive, this is true. So, angle C is obtuse.But I don't know if that helps me directly.Wait, maybe I can find the value of k.Since E is on AC, and EC is vertical, meaning that E is at (x,y) where x is the same as F on BC, which is (x,0). So, E is at (x,y), and F is at (x,0). Since E is on AC, which goes from (-d,e) to (0,0), the parametric equation is x = -d + t*d, y = e - t*e.So, E is at (x,y) = (-d + t*d, e - t*e). Then, F is at (x,0) = (-d + t*d, 0).Now, the length of EC is y = e - t*e.But EC is also the vertical distance from E to BC, which is y.Now, I need to find t such that E is on AC and EC is vertical. But as above, any E on AC will have EC vertical if E is projected down to BC. Wait, no, only if E is directly above F on BC.Wait, I think I'm stuck here. Maybe I can use similar triangles.Since MD is perpendicular to BC, and EC is perpendicular to BC, both MD and EC are heights onto BC. Maybe triangles MBD and EBC are similar.Wait, triangle MBD: points M, B, D.Triangle EBC: points E, B, C.Are they similar?Let me check the angles.Angle at B is common to both triangles.Angle at D is right angle, angle at C is right angle.So, triangles MBD and EBC are similar by AA similarity.Therefore, the ratio of sides is the same.So, MB/EB = BD/BC = MD/EC.Since M is the midpoint of AB, MB = AB/2.Let me denote AB as c, so MB = c/2.Let me denote BC as a, so BC = a.Let me denote BD as x, so BD = x.Then, from similarity:MB/EB = BD/BC => (c/2)/EB = x/a => EB = (c/2)*a/x.Also, MD/EC = BD/BC => MD/EC = x/a => MD = (x/a)*EC.But I need to find the area of triangle BED.Wait, triangle BED has base BD = x and height EC.So, area of BED is (1/2)*x*EC.From above, EC = (a/x)*MD.But MD is the height from M to BC, which is the y-coordinate of M, which is e/2.Wait, in my coordinate system, M is at ((c - d)/2, e/2), so MD is e/2.Therefore, EC = (a/x)*(e/2).So, area of BED is (1/2)*x*(a/x)*(e/2) = (1/2)*(a*e)/2 = (a*e)/4.But from the area of ABC, which is (1/2)*a*e = 30, so a*e = 60.Therefore, area of BED is 60/4 = 15.So, the answer is 15, which is option C.Wait, let me verify this.From the similarity of triangles MBD and EBC, we have:MB/EB = BD/BC = MD/EC.We found that area of BED is (1/2)*BD*EC = (1/2)*x*EC.From similarity, EC = (a/x)*MD, and MD = e/2.So, EC = (a/x)*(e/2).Thus, area of BED = (1/2)*x*(a/x)*(e/2) = (1/2)*(a*e)/2 = (a*e)/4.Since a*e = 60, area of BED is 60/4 = 15.Yes, that makes sense.

💡First, I need to determine the total distance from the dormitory to the city. Let's denote this distance as ( D ) kilometers.According to the problem, the student traveled:- ( frac{1}{3} ) of the way by foot,- ( frac{3}{5} ) of the way by bus, and- the remaining 2 kilometers by car.I can set up the equation:[frac{1}{3}D + frac{3}{5}D + 2 = D]To solve for ( D ), I'll find a common denominator for the fractions, which is 15:[frac{5}{15}D + frac{9}{15}D + 2 = D]Combining the fractions:[frac{14}{15}D + 2 = D]Subtracting ( frac{14}{15}D ) from both sides:[2 = D - frac{14}{15}D][2 = frac{1}{15}D]Finally, multiplying both sides by 15 to solve for ( D ):[D = 30]Therefore, the distance from the dormitory to the city is 30 kilometers.

💡Alright, so I have this problem here: I need to find the value of ( m ) such that the coefficient of ( x^3 y^3 ) in the expansion of ( (mx + y)^6 ) is ( -160 ). Hmm, okay, let me think about how to approach this.First off, I remember that when you expand a binomial like ( (a + b)^n ), you can use the binomial theorem. The binomial theorem says that each term in the expansion is of the form ( binom{n}{k} a^{n - k} b^k ), right? So, in this case, ( a ) would be ( mx ) and ( b ) would be ( y ), and ( n ) is 6.I need to find the coefficient of the term ( x^3 y^3 ). That means I need to find the term where the power of ( x ) is 3 and the power of ( y ) is also 3. So, let me figure out what ( k ) would be for that term.Since ( b ) is ( y ) and it's raised to the power of ( k ), and I want ( y^3 ), that means ( k = 3 ). Therefore, the term corresponding to ( k = 3 ) will be ( binom{6}{3} (mx)^{6 - 3} y^3 ). Simplifying that, it becomes ( binom{6}{3} (mx)^3 y^3 ).Now, let me compute ( binom{6}{3} ). I know that ( binom{n}{k} = frac{n!}{k!(n - k)!} ). So, plugging in the numbers, ( binom{6}{3} = frac{6!}{3!3!} ). Calculating that, ( 6! = 720 ), ( 3! = 6 ), so it's ( frac{720}{6 times 6} = frac{720}{36} = 20 ). Okay, so the binomial coefficient is 20.Next, let's look at ( (mx)^3 ). That would be ( m^3 x^3 ). So, putting it all together, the term is ( 20 times m^3 x^3 y^3 ). Therefore, the coefficient of ( x^3 y^3 ) is ( 20 m^3 ).According to the problem, this coefficient is equal to ( -160 ). So, I can set up the equation:[ 20 m^3 = -160 ]Now, I need to solve for ( m ). Let's divide both sides by 20 to isolate ( m^3 ):[ m^3 = frac{-160}{20} ][ m^3 = -8 ]To find ( m ), I take the cube root of both sides:[ m = sqrt[3]{-8} ]I know that the cube root of -8 is -2 because ( (-2)^3 = -8 ). So, ( m = -2 ).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the binomial expansion, I identified the correct term, calculated the binomial coefficient correctly, substituted ( k = 3 ), and then set up the equation with the given coefficient. Solving for ( m ) gave me -2, which seems right because when I cube -2, I get -8, and multiplying by 20 gives -160. Yeah, that checks out.I think I've got it. The value of ( m ) is -2.**Final Answer**The value of ( m ) is boxed{-2}.

💡Okay, so I have this problem here: "m=2" is a condition for the function f(x) = -3 + mx + x² to have two distinct roots. I need to figure out whether this condition is sufficient, necessary, both, or neither. The options are A, B, C, D as given.First, let me recall what it means for a quadratic function to have two distinct roots. A quadratic equation ax² + bx + c = 0 has two distinct real roots if its discriminant is positive. The discriminant D is given by D = b² - 4ac. So, if D > 0, there are two distinct real roots.In this case, the function is f(x) = x² + mx - 3. So, comparing to the standard quadratic equation, a = 1, b = m, and c = -3. Therefore, the discriminant D would be m² - 4*1*(-3) = m² + 12.Wait, that's interesting. So D = m² + 12. Since m² is always non-negative, D is always greater than or equal to 12, right? Because m² is at least 0, so m² + 12 is at least 12. Therefore, D is always positive, regardless of the value of m.Hold on, that means that for any real number m, the quadratic function f(x) = x² + mx - 3 will always have two distinct real roots because the discriminant is always positive. So, regardless of what m is, as long as it's a real number, the function will have two distinct roots.But the question is specifically about the condition m=2. So, is m=2 a necessary condition, a sufficient condition, both, or neither?Let me think about what necessary and sufficient conditions mean. A necessary condition is one that must be true for the statement to hold, but it might not be enough on its own. A sufficient condition is one that, if it is true, guarantees the statement is true, but the statement might be true even if the condition isn't met.In this case, since the discriminant is always positive for any real m, the function will always have two distinct roots, no matter what m is. So, m=2 is just one specific value of m that makes the discriminant positive. But since any m would work, m=2 isn't necessary because the function can have two distinct roots even if m isn't 2. However, m=2 is sufficient because setting m=2 does ensure that the discriminant is positive, hence two distinct roots.Wait, but let me double-check that. If m=2, then the discriminant D = 2² + 12 = 4 + 12 = 16, which is positive, so yes, two distinct roots. But if m is, say, 3, then D = 9 + 12 = 21, which is also positive. Similarly, if m is 0, D = 0 + 12 = 12, which is still positive. So, no matter what m is, D is always positive.Therefore, m=2 is just one example where the discriminant is positive, but it's not the only one. So, m=2 is sufficient because it guarantees two distinct roots, but it's not necessary because two distinct roots can occur even when m isn't 2.So, putting it all together, the condition m=2 is sufficient but not necessary. Therefore, the answer should be option A.But wait, let me make sure I didn't make a mistake. Is there any case where m=2 is necessary? For instance, is there a scenario where m has to be 2 for the function to have two distinct roots? But no, as I saw earlier, any m would work because D is always positive. So, m=2 isn't required; it's just one possible value.Therefore, I think my conclusion is correct. m=2 is a sufficient condition but not a necessary one. So, the answer is A.**Final Answer**boxed{A}

💡To determine which expression is equal to ( x^{3n+1} ), I will evaluate each option step by step.**Option A: ( (x^{3})^{n+1} )**- Using the power of a power property, ( (x^{a})^{b} = x^{ab} ), this becomes ( x^{3(n+1)} = x^{3n+3} ).- This is not equal to ( x^{3n+1} ).**Option B: ( (x^{n+1})^{3} )**- Again, using the power of a power property, this becomes ( x^{(n+1) cdot 3} = x^{3n+3} ).- This is also not equal to ( x^{3n+1} ).**Option C: ( x^{3} cdot x^{n} cdot x )**- Using the product of powers property, ( x^{a} cdot x^{b} = x^{a+b} ), this simplifies to ( x^{3+n+1} = x^{n+4} ).- This does not match ( x^{3n+1} ).**Option D: ( x cdot x^{3n} )**- Applying the product of powers property, this becomes ( x^{1} cdot x^{3n} = x^{3n+1} ).- This is exactly equal to ( x^{3n+1} ).After evaluating all options, only Option D matches the given expression.