💡Okay, so I have this problem: I need to find the largest value of ( n ) such that the quadratic expression ( 7x^2 + nx + 56 ) can be factored into the product of two linear factors with integer coefficients. Hmm, let me think about how to approach this.First, I remember that factoring a quadratic expression generally involves breaking it down into two binomials. Since the coefficient of ( x^2 ) is 7, which is a prime number, the factors of the quadratic must be of the form ( (7x + a)(x + b) ), where ( a ) and ( b ) are integers. That makes sense because when you multiply ( 7x ) by ( x ), you get ( 7x^2 ), which matches the original expression.Now, if I expand ( (7x + a)(x + b) ), I should get ( 7x^2 + (7b + a)x + ab ). Comparing this to the original expression ( 7x^2 + nx + 56 ), I can set up the following equations:1. The constant term: ( ab = 56 )2. The middle term: ( 7b + a = n )So, my goal is to find integers ( a ) and ( b ) such that their product is 56, and then calculate ( n ) as ( 7b + a ). I need to find all possible pairs of integers ( (a, b) ) that multiply to 56 and then determine which pair gives the largest value of ( n ).Let me list all the pairs of integers whose product is 56. Since 56 is positive, both ( a ) and ( b ) can be either both positive or both negative. However, since we're looking for the largest possible ( n ), which is ( 7b + a ), I think positive integers will give a larger result because adding positive numbers will lead to a larger sum. So, I'll focus on positive pairs first.The positive integer pairs that multiply to 56 are:- ( (1, 56) )- ( (2, 28) )- ( (4, 14) )- ( (7, 8) )Wait, but actually, these are all the positive factor pairs. I should also consider the reverse of each pair because ( a ) and ( b ) can switch places, which might affect the value of ( n ).So, the complete list of positive factor pairs is:- ( (1, 56) )- ( (56, 1) )- ( (2, 28) )- ( (28, 2) )- ( (4, 14) )- ( (14, 4) )- ( (7, 8) )- ( (8, 7) )Now, for each of these pairs, I'll calculate ( n = 7b + a ) and see which one gives the largest value.Starting with ( (1, 56) ):- ( a = 1 ), ( b = 56 )- ( n = 7*56 + 1 = 392 + 1 = 393 )Next, ( (56, 1) ):- ( a = 56 ), ( b = 1 )- ( n = 7*1 + 56 = 7 + 56 = 63 )Moving on to ( (2, 28) ):- ( a = 2 ), ( b = 28 )- ( n = 7*28 + 2 = 196 + 2 = 198 )Then, ( (28, 2) ):- ( a = 28 ), ( b = 2 )- ( n = 7*2 + 28 = 14 + 28 = 42 )Next, ( (4, 14) ):- ( a = 4 ), ( b = 14 )- ( n = 7*14 + 4 = 98 + 4 = 102 )Then, ( (14, 4) ):- ( a = 14 ), ( b = 4 )- ( n = 7*4 + 14 = 28 + 14 = 42 )Now, ( (7, 8) ):- ( a = 7 ), ( b = 8 )- ( n = 7*8 + 7 = 56 + 7 = 63 )Lastly, ( (8, 7) ):- ( a = 8 ), ( b = 7 )- ( n = 7*7 + 8 = 49 + 8 = 57 )Okay, so compiling all these ( n ) values:- 393, 63, 198, 42, 102, 42, 63, 57Looking at these, the largest value is 393. So, that must be the maximum possible ( n ) for which the quadratic can be factored into two linear terms with integer coefficients.But wait, just to make sure I haven't missed any factor pairs, let me double-check. 56 is 7 times 8, and its factors are 1, 2, 4, 7, 8, 14, 28, 56. So, I think I have all the positive factor pairs. Also, considering negative factors might lead to negative ( n ), but since we're looking for the largest ( n ), which is positive, I don't need to consider negative pairs.Therefore, I'm confident that the largest value of ( n ) is 393.

💡Alright, so I've got this problem here: "How much does 10 moles of a certain compound weigh if the total weight is 1180 grams?" And there's a hint that the compound is H2CrO4. Okay, let's see how to tackle this.First off, I remember that when dealing with moles and weight, molar mass is key. Molar mass is the mass of one mole of a substance, and it's calculated by adding up the atomic masses of all the atoms in the molecule. So, for H2CrO4, I need to find the molar mass by looking at the atomic masses of hydrogen (H), chromium (Cr), and oxygen (O).Let me recall the atomic masses:- Hydrogen (H) is about 1.01 g/mol.- Chromium (Cr) is around 52.00 g/mol.- Oxygen (O) is approximately 16.00 g/mol.Now, H2CrO4 has 2 hydrogen atoms, 1 chromium atom, and 4 oxygen atoms. So, calculating the molar mass:- Hydrogen: 2 * 1.01 g/mol = 2.02 g/mol- Chromium: 1 * 52.00 g/mol = 52.00 g/mol- Oxygen: 4 * 16.00 g/mol = 64.00 g/molAdding these up: 2.02 + 52.00 + 64.00 = 118.02 g/mol.Okay, so one mole of H2CrO4 weighs approximately 118.02 grams. Now, the question is asking about 10 moles. So, I need to multiply the molar mass by 10 to find the total weight.10 moles * 118.02 g/mol = 1180.2 grams.Wait a minute, the problem states that the total weight is 1180 grams. My calculation gives 1180.2 grams, which is very close but not exactly 1180 grams. Is there something I'm missing here?Maybe the slight difference is due to rounding off the atomic masses. Let me double-check the atomic masses with more precise values:- Hydrogen (H): 1.008 g/mol- Chromium (Cr): 52.00 g/mol- Oxygen (O): 16.00 g/molRecalculating:- Hydrogen: 2 * 1.008 = 2.016 g/mol- Chromium: 52.00 g/mol- Oxygen: 4 * 16.00 = 64.00 g/molAdding them up: 2.016 + 52.00 + 64.00 = 118.016 g/mol.Now, 10 moles * 118.016 g/mol = 1180.16 grams.Still, it's approximately 1180 grams when rounded to the nearest whole number. So, it seems that 10 moles of H2CrO4 weigh about 1180 grams, which matches the given total weight.I think I've got it figured out. The slight discrepancy is just due to rounding the atomic masses, and the answer aligns with the total weight provided.

💡Alright, so I have this problem about a function f(x) = 2 sin(ωx + φ), where ω is positive and φ is between 0 and π/2. The problem gives me some conditions: f(x₁) = 2, f(x₂) = 0, and the minimum value of |x₁ - x₂| is 1/2. Also, f(1/2) = 1. I need to find the interval where f(x) is monotonically increasing, and then choose the correct option from A to D.First, I need to understand what each condition means. The function is a sine wave with amplitude 2, frequency ω, and phase shift φ. The maximum value of the sine function is 1, so multiplying by 2 makes the maximum 2. Similarly, the minimum is -2.Given that f(x₁) = 2, that means at x = x₁, the sine function reaches its maximum. So, sin(ωx₁ + φ) = 1. Similarly, f(x₂) = 0 means that at x = x₂, the sine function is 0. So, sin(ωx₂ + φ) = 0.The minimum value of |x₁ - x₂| is 1/2. This suggests that the distance between a maximum and a zero crossing is 1/2. In a sine wave, the distance between a maximum and the next zero crossing is a quarter of the period. So, if the period is T, then T/4 = 1/2. Therefore, T = 2.The period T of a sine function is given by T = 2π / ω. So, if T = 2, then ω = 2π / T = 2π / 2 = π. So, ω is π.Now, the function is f(x) = 2 sin(πx + φ). Next, we have f(1/2) = 1. Let's plug x = 1/2 into the function:f(1/2) = 2 sin(π*(1/2) + φ) = 2 sin(π/2 + φ) = 1.So, sin(π/2 + φ) = 1/2.We know that sin(π/2 + φ) = cos(φ). So, cos(φ) = 1/2.Since φ is between 0 and π/2, cos(φ) = 1/2 implies that φ = π/3. Because cos(π/3) = 1/2.So, now we have the function f(x) = 2 sin(πx + π/3).Next, we need to find the intervals where f(x) is monotonically increasing. For a sine function, it's increasing where its derivative is positive.The derivative of f(x) is f’(x) = 2π cos(πx + π/3).We need to find where f’(x) > 0, which is where cos(πx + π/3) > 0.So, when is cos(θ) > 0? Cosine is positive in the intervals (-π/2 + 2πk, π/2 + 2πk) for any integer k.So, we set up the inequality:-π/2 + 2πk < πx + π/3 < π/2 + 2πk.Let's solve for x.First, subtract π/3 from all parts:-π/2 - π/3 + 2πk < πx < π/2 - π/3 + 2πk.Compute the constants:-π/2 - π/3 = -(3π/6 + 2π/6) = -5π/6.Similarly, π/2 - π/3 = (3π/6 - 2π/6) = π/6.So, the inequality becomes:-5π/6 + 2πk < πx < π/6 + 2πk.Divide all parts by π:-5/6 + 2k < x < 1/6 + 2k.So, the function f(x) is increasing on the intervals (-5/6 + 2k, 1/6 + 2k) for any integer k.Looking at the options, option B is [-5/6 + 2k, 1/6 + 2k], which matches our result. So, the answer should be B.Wait, but let me double-check. The derivative is positive when cosine is positive, which is in those intervals. So, yes, the function is increasing on those intervals. So, I think B is correct.**Final Answer**boxed{B}

💡Alright, so I need to find the center of a given circle using a drafting triangle. Hmm, okay, let's see. I remember that a drafting triangle is a right-angled tool, usually with angles of 90 degrees, 45 degrees, and 45 degrees, or sometimes 30 degrees and 60 degrees. But in this case, since it's a drafting triangle, I think it's the right-angled one, the 90-degree one.Okay, so the circle is given, and I need to find its center. I recall that the center of a circle is the point equidistant from all points on the circumference. So, if I can find two points on the circle that are directly opposite each other, the line connecting them would be a diameter, and the midpoint of that diameter would be the center.But how do I find such points with a drafting triangle? Maybe I can use the right angle of the triangle somehow. Wait, I think there's a property related to right angles and circles. If I have a right-angled triangle inscribed in a circle, the hypotenuse is the diameter of the circle. Is that correct? Let me think.Yes, I remember something like that from geometry. If you have a triangle inscribed in a circle where one of the angles is a right angle, then the side opposite that right angle is the diameter of the circle. So, if I can create a right-angled triangle with the vertex of the right angle on the circumference of the circle, then the hypotenuse would be a diameter.Alright, so here's what I can do: place the right angle of the drafting triangle on a point on the circumference of the circle. Then, align the two legs of the triangle so that they intersect the circle at two other points. The line connecting these two intersection points should be a diameter.But wait, how do I ensure that the legs of the triangle intersect the circle? Maybe I need to adjust the position of the triangle so that both legs touch the circle. Once I have that, the line between those two points is a diameter, and its midpoint is the center.But I need to confirm this. Let me visualize it. If I have a circle, and I place the right angle of the triangle on the circumference, then the two legs extend outwards. If I can position the triangle such that both legs intersect the circle again, then those two points should be endpoints of a diameter.Okay, so I'll try that. I'll take the drafting triangle, place the right angle vertex on the circle, and then adjust the triangle so that the other two sides intersect the circle at two distinct points. Let's call these points A and B. Then, the line AB should be a diameter.Now, to find the center, I need to find the midpoint of AB. But how do I find the midpoint with a drafting triangle? Maybe I can use the triangle to draw a perpendicular bisector of AB. If I can construct a perpendicular line to AB that passes through its midpoint, the intersection point of this perpendicular bisector with AB would be the center.Alternatively, if I can create another diameter using the same method, the intersection point of the two diameters would be the center. That might be easier. So, I can repeat the process: place the right angle of the triangle on another point on the circumference, align the legs to intersect the circle at two new points, C and D, and then draw the diameter CD. The point where AB and CD intersect would be the center of the circle.But I need to make sure that AB and CD are indeed diameters. How can I verify that? Well, since I used the right angle property, and by the inscribed angle theorem, any triangle inscribed in a circle with a right angle must have its hypotenuse as the diameter. So, as long as I correctly positioned the triangle each time, AB and CD should both be diameters.Okay, so to summarize my steps:1. Place the right angle vertex of the drafting triangle on a point on the circumference of the circle.2. Adjust the triangle so that the two legs intersect the circle at two other points, A and B.3. Draw the line AB, which should be a diameter.4. Repeat the process by placing the right angle vertex on another point on the circumference and drawing another diameter CD.5. The intersection point of AB and CD is the center of the circle.But wait, what if I can't get the legs of the triangle to intersect the circle? Maybe the circle is too small or the triangle is too large. I need to ensure that the triangle is appropriately sized for the given circle. If the circle is too small, I might need a smaller drafting triangle or adjust my technique.Also, I should consider that the drafting triangle might not be perfectly aligned with the circle's center initially. So, I might need to make several attempts to get accurate diameters. Maybe drawing multiple diameters and finding their intersection would give a more precise center.Another thought: if I only draw one diameter, I can find its midpoint by folding the paper or using a ruler to measure halfway. But since I'm using a drafting triangle, perhaps I can use it to draw the perpendicular bisector.Wait, how would I draw a perpendicular bisector with a drafting triangle? I can align one leg along the diameter AB, then use the other leg to draw a line perpendicular to AB at its midpoint. The point where this perpendicular line intersects AB is the center.But I'm not sure if I can accurately find the midpoint without a ruler. Maybe I need to estimate it by eye or use some other method. Alternatively, using two diameters as I initially thought might be more straightforward.Let me think about potential errors. If I don't place the right angle vertex exactly on the circumference, the resulting line might not be a diameter. Similarly, if the triangle is not aligned properly, the legs might not intersect the circle symmetrically. So, precision is key here.Also, if the circle is not perfectly round, this method might not work as intended. But assuming the circle is perfect, this method should find the center accurately.In summary, the process involves using the right angle of the drafting triangle to inscribe a right-angled triangle in the circle, thereby identifying diameters, and then finding their intersection point, which is the center.I think I've got a good plan. Now, let's try to outline the steps clearly.

💡Alright, so I have this problem where I need to find the minimum value of |AM| + |MF|, where M is a moving point on the parabola y² = 8x, F is the focus of the parabola, and A is a point on the circle C: (x - 3)² + (y + 1)² = 1. Hmm, okay, let me try to break this down step by step.First, I remember that for a parabola in the form y² = 4ax, the focus is at (a, 0). In this case, the equation is y² = 8x, so comparing it to y² = 4ax, I see that 4a = 8, which means a = 2. So, the focus F should be at (2, 0). Got that down.Next, the problem mentions a point A lying on the circle (x - 3)² + (y + 1)² = 1. This is a circle with center at (3, -1) and radius 1. So, point A can be anywhere on the circumference of this circle.Now, I need to find the minimum value of |AM| + |MF|, where M is on the parabola. Hmm, okay. So, this is like finding the shortest path from A to M to F, where M is constrained to lie on the parabola.I recall that in problems involving minimizing the sum of distances, reflection properties can sometimes be useful. For example, in the case of ellipses, the sum of distances from two foci is constant, but here we have a parabola and a circle.Wait, the parabola has a focus and a directrix. Maybe I can use the definition of a parabola here. The definition says that any point on the parabola is equidistant from the focus and the directrix. So, for any point M on the parabola, |MF| is equal to the distance from M to the directrix.What's the directrix of the parabola y² = 8x? Since the standard form is y² = 4ax, the directrix is x = -a. Here, a = 2, so the directrix is x = -2.So, for any point M on the parabola, |MF| is equal to the horizontal distance from M to the line x = -2. That is, if M has coordinates (x, y), then |MF| = x + 2.Wait, is that right? Let me think. If M is on the parabola, then its distance to F is equal to its distance to the directrix. The distance to F is sqrt[(x - 2)² + y²], and the distance to the directrix is |x - (-2)| = |x + 2|. Since M is on the parabola, these two distances are equal. So, sqrt[(x - 2)² + y²] = |x + 2|. Squaring both sides, (x - 2)² + y² = (x + 2)². Expanding both sides: x² - 4x + 4 + y² = x² + 4x + 4. Simplifying, -4x + y² = 4x, so y² = 8x, which matches the given equation. Okay, so that's correct.So, |MF| = x + 2, where x is the x-coordinate of M. Therefore, |MF| = x + 2.Now, I need to find |AM| + |MF|. Since |MF| = x + 2, this becomes |AM| + x + 2. Hmm, but I'm not sure if that helps directly. Maybe I should think about |AM| + |MF| in terms of geometry.Alternatively, since |MF| is equal to the distance from M to the directrix, which is a vertical line x = -2, maybe I can use some reflection property here. In some problems, reflecting a point across a line can help find minimal paths.Let me consider reflecting point F across the directrix. Wait, the directrix is x = -2, and F is at (2, 0). Reflecting F over the directrix would give a point F' such that the directrix is the perpendicular bisector of the segment FF'. So, the reflection of F(2, 0) over x = -2 would be F'(-6, 0). Because the distance from F to the directrix is 4 units (from x=2 to x=-2), so reflecting it would place F' 4 units on the other side, at x = -6.But I'm not sure if that helps here. Maybe I should think about reflecting point A instead.Wait, another approach: since |MF| is equal to the distance from M to the directrix, which is x = -2, then |MF| = x + 2, as we established earlier. So, |AM| + |MF| = |AM| + x + 2.But x is the x-coordinate of M, which is on the parabola y² = 8x. So, x is related to y by x = y² / 8.Hmm, maybe I can express |AM| in terms of coordinates. Let me denote point M as (y² / 8, y). Then, point A is on the circle (x - 3)² + (y + 1)² = 1. So, point A can be represented as (3 + cosθ, -1 + sinθ), where θ is a parameter between 0 and 2π.So, |AM| is the distance between (y² / 8, y) and (3 + cosθ, -1 + sinθ). That would be sqrt[(y² / 8 - 3 - cosθ)² + (y + 1 - sinθ)²].And |MF| is x + 2, which is (y² / 8) + 2.So, |AM| + |MF| becomes sqrt[(y² / 8 - 3 - cosθ)² + (y + 1 - sinθ)²] + (y² / 8) + 2.This seems complicated. Maybe there's a better way to approach this.Wait, perhaps instead of parameterizing A, I can think about the minimal value of |AM| + |MF| over all M on the parabola and A on the circle.Alternatively, since A is on the circle, maybe I can consider the minimal |AM| + |MF| as the minimal distance from A to M plus the minimal distance from M to F, but I'm not sure if that's the right way to think about it.Wait, another idea: since |MF| is equal to the distance from M to the directrix, which is x = -2, maybe I can think of |AM| + |MF| as |AM| + distance from M to directrix.But I'm not sure how that helps. Maybe I can use the triangle inequality or something.Alternatively, perhaps I can fix M and find the minimal |AM| over A on the circle, then add |MF|, and then minimize over M.So, for a fixed M, the minimal |AM| would be the distance from M to the center of the circle minus the radius. The center of the circle is (3, -1), radius 1.So, |AM| ≥ |MC| - 1, where C is (3, -1).Therefore, |AM| + |MF| ≥ |MC| - 1 + |MF|.So, if I can find the minimal value of |MC| + |MF| - 1, that would give me the minimal |AM| + |MF|.But |MC| + |MF| is the sum of distances from M to C and from M to F. Hmm, I wonder if there's a way to minimize this sum.Wait, in general, the minimal sum of distances from a point to two fixed points is achieved when the point lies on the line segment connecting those two points. But here, M is constrained to lie on the parabola.So, perhaps the minimal |MC| + |MF| occurs when M is the intersection point of the parabola and the line segment connecting C and F.Let me check: C is at (3, -1), F is at (2, 0). So, the line segment from C to F goes from (3, -1) to (2, 0). Let me find the equation of this line.The slope is (0 - (-1))/(2 - 3) = 1 / (-1) = -1. So, the equation is y - (-1) = -1(x - 3), which simplifies to y + 1 = -x + 3, so y = -x + 2.Now, find the intersection of this line with the parabola y² = 8x.Substitute y = -x + 2 into y² = 8x:(-x + 2)² = 8xx² - 4x + 4 = 8xx² - 12x + 4 = 0Using quadratic formula: x = [12 ± sqrt(144 - 16)] / 2 = [12 ± sqrt(128)] / 2 = [12 ± 8√2] / 2 = 6 ± 4√2.So, x = 6 + 4√2 or x = 6 - 4√2.Since the parabola y² = 8x opens to the right, x must be non-negative. Both 6 + 4√2 and 6 - 4√2 are positive, since 4√2 ≈ 5.656, so 6 - 5.656 ≈ 0.344, which is positive.So, the intersection points are at x = 6 + 4√2 and x = 6 - 4√2.Corresponding y-values: y = -x + 2.For x = 6 + 4√2: y = -(6 + 4√2) + 2 = -4 - 4√2.For x = 6 - 4√2: y = -(6 - 4√2) + 2 = -4 + 4√2.So, the points of intersection are (6 + 4√2, -4 - 4√2) and (6 - 4√2, -4 + 4√2).Now, let's check if these points lie on the parabola and on the line segment CF.Wait, the line segment from C(3, -1) to F(2, 0) is from (3, -1) to (2, 0). The points we found are (6 + 4√2, -4 - 4√2) and (6 - 4√2, -4 + 4√2). These points are far from the segment CF, so they are not on the segment but on the extended line.Therefore, the minimal |MC| + |MF| might not occur at these points. Hmm, maybe I need a different approach.Wait, perhaps I can use the reflection property of the parabola. Since |MF| is equal to the distance from M to the directrix, which is x = -2, maybe I can reflect point A over the directrix and then find the distance from the reflection to F.Wait, let me think. If I reflect A over the directrix x = -2, I get a point A'. Then, |AM| = |A'M| because reflection preserves distances. So, |AM| + |MF| = |A'M| + |MF|. But since |MF| is equal to the distance from M to the directrix, which is the same as the distance from M to F, I'm not sure if that helps.Wait, no, |MF| is equal to the distance from M to the directrix, which is x = -2. So, |MF| = x + 2, where x is the x-coordinate of M.Alternatively, maybe I can think of |AM| + |MF| as |AM| + (x + 2). Since x = y² / 8, this becomes |AM| + y² / 8 + 2.But I'm not sure if that helps either.Wait, another idea: since |MF| is equal to the distance from M to the directrix, which is x = -2, maybe I can consider the point A reflected over the directrix, say A'', and then |AM| + |MF| would be equal to |AM| + |MN|, where N is the foot of the perpendicular from M to the directrix.But since |MF| = |MN|, then |AM| + |MF| = |AM| + |MN|. Now, if I can find a way to minimize |AM| + |MN|, that would give me the minimal value.But I'm not sure how to proceed from here. Maybe I can consider the point A'' as the reflection of A over the directrix, and then |AM| + |MN| would be equal to |AM| + |A''M|, but I'm not sure.Wait, actually, reflecting A over the directrix might not be straightforward because the directrix is a vertical line. So, reflecting A over x = -2 would change the x-coordinate. If A is (a, b), then A'' would be (-4 - a, b).But I'm not sure if that helps. Maybe I can consider the minimal |AM| + |MF| as the minimal |AM| + |MN|, and since N is on the directrix, maybe I can find the minimal distance from A to N via M on the parabola.Wait, this is getting a bit convoluted. Maybe I should try a different approach.Let me consider the problem again: minimize |AM| + |MF|, where M is on the parabola y² = 8x, F is (2, 0), and A is on the circle (x - 3)² + (y + 1)² = 1.I need to find the minimal value of this sum. Maybe I can fix M and find the minimal |AM|, then add |MF|, and then minimize over M.As I thought earlier, for a fixed M, the minimal |AM| is |MC| - 1, where C is (3, -1). So, |AM| + |MF| ≥ |MC| - 1 + |MF|.Therefore, the minimal value would be the minimal value of |MC| + |MF| - 1.So, I need to find the minimal value of |MC| + |MF|, where M is on the parabola, and then subtract 1.Now, |MC| + |MF| is the sum of distances from M to C and from M to F. To minimize this sum, I can think of it as finding a point M on the parabola such that the sum of distances from M to C and M to F is minimized.This is similar to the Fermat-Torricelli problem, where we need to find a point minimizing the sum of distances to given points, but here M is constrained to lie on the parabola.One way to approach this is to use calculus. Let me parameterize M on the parabola. Since the parabola is y² = 8x, I can write M as (2t², 4t), where t is a parameter. This is because for y² = 8x, x = y² / 8, so if I let y = 4t, then x = (16t²) / 8 = 2t².So, M = (2t², 4t).Now, point C is (3, -1), and F is (2, 0).Compute |MC|: distance between (2t², 4t) and (3, -1):|MC| = sqrt[(2t² - 3)² + (4t + 1)²]Similarly, |MF| = sqrt[(2t² - 2)² + (4t - 0)²] = sqrt[(2t² - 2)² + (4t)²]So, |MC| + |MF| = sqrt[(2t² - 3)² + (4t + 1)²] + sqrt[(2t² - 2)² + (4t)²]This is a function of t, and I need to find its minimum.This seems complicated, but maybe I can find the derivative and set it to zero to find critical points.Let me denote f(t) = sqrt[(2t² - 3)² + (4t + 1)²] + sqrt[(2t² - 2)² + (4t)²]Compute f'(t):f'(t) = [ (4t)(2t² - 3) + 4(4t + 1) ] / sqrt[(2t² - 3)² + (4t + 1)²] + [ (4t)(2t² - 2) + 4(4t) ] / sqrt[(2t² - 2)² + (4t)²]Simplify numerator of first term:4t(2t² - 3) + 4(4t + 1) = 8t³ - 12t + 16t + 4 = 8t³ + 4t + 4Denominator is sqrt[(2t² - 3)² + (4t + 1)²]Numerator of second term:4t(2t² - 2) + 16t = 8t³ - 8t + 16t = 8t³ + 8tDenominator is sqrt[(2t² - 2)² + (4t)²]So, f'(t) = [8t³ + 4t + 4] / sqrt[(2t² - 3)² + (4t + 1)²] + [8t³ + 8t] / sqrt[(2t² - 2)² + (4t)²]Set f'(t) = 0:[8t³ + 4t + 4] / sqrt[(2t² - 3)² + (4t + 1)²] + [8t³ + 8t] / sqrt[(2t² - 2)² + (4t)²] = 0This equation looks quite complicated. Maybe there's a symmetry or a substitution that can simplify it.Alternatively, perhaps I can assume that the minimal occurs when M lies on the line connecting C and F. Earlier, I found that the line CF has the equation y = -x + 2, and it intersects the parabola at (6 ± 4√2, -4 ∓ 4√2). But these points are far from the segment CF, so maybe the minimal occurs somewhere else.Alternatively, perhaps the minimal occurs when M is such that the angles made by MC and MF with the tangent at M are equal, satisfying some reflection property. But I'm not sure.Wait, another idea: since |MF| = x + 2, and x = 2t², so |MF| = 2t² + 2.And |MC| = sqrt[(2t² - 3)² + (4t + 1)²]So, f(t) = sqrt[(2t² - 3)² + (4t + 1)²] + 2t² + 2Wait, no, that's not correct because |MF| is sqrt[(2t² - 2)² + (4t)²], which simplifies to sqrt[4t⁴ - 8t² + 4 + 16t²] = sqrt[4t⁴ + 8t² + 4] = sqrt[4(t⁴ + 2t² + 1)] = 2(t² + 1). So, |MF| = 2(t² + 1).Wait, that's interesting. So, |MF| = 2(t² + 1).Therefore, f(t) = sqrt[(2t² - 3)² + (4t + 1)²] + 2(t² + 1)So, f(t) = sqrt[(2t² - 3)² + (4t + 1)²] + 2t² + 2Now, let's compute the derivative f'(t):f'(t) = [ (4t)(2t² - 3) + 4(4t + 1) ] / sqrt[(2t² - 3)² + (4t + 1)²] + 4tSimplify numerator:4t(2t² - 3) + 4(4t + 1) = 8t³ - 12t + 16t + 4 = 8t³ + 4t + 4So, f'(t) = (8t³ + 4t + 4) / sqrt[(2t² - 3)² + (4t + 1)²] + 4tSet f'(t) = 0:(8t³ + 4t + 4) / sqrt[(2t² - 3)² + (4t + 1)²] + 4t = 0This still looks complicated. Maybe I can try to find t such that this equation holds.Alternatively, perhaps I can make a substitution to simplify the expression. Let me denote u = t², so t = sqrt(u). But I'm not sure if that helps.Alternatively, maybe I can square both sides to eliminate the square root, but that might complicate things further.Wait, perhaps I can consider specific values of t to see if I can find a minimum.Let me try t = 0:f(0) = sqrt[(0 - 3)² + (0 + 1)²] + 2(0 + 1) = sqrt[9 + 1] + 2 = sqrt(10) + 2 ≈ 3.16 + 2 = 5.16f'(0) = (0 + 0 + 4)/sqrt[9 + 1] + 0 = 4 / sqrt(10) ≈ 1.264 > 0So, the function is increasing at t=0.Try t = 1:f(1) = sqrt[(2 - 3)² + (4 + 1)²] + 2(1 + 1) = sqrt[1 + 25] + 4 = sqrt(26) + 4 ≈ 5.1 + 4 = 9.1f'(1) = (8 + 4 + 4)/sqrt[1 + 25] + 4 = 16 / sqrt(26) + 4 ≈ 3.11 + 4 = 7.11 > 0Still increasing.Try t = -1:f(-1) = sqrt[(2 - 3)² + (-4 + 1)²] + 2(1 + 1) = sqrt[1 + 9] + 4 = sqrt(10) + 4 ≈ 3.16 + 4 = 7.16f'(-1) = (-8 - 4 + 4)/sqrt[1 + 9] + (-4) = (-8)/sqrt(10) - 4 ≈ -2.53 - 4 = -6.53 < 0So, at t = -1, the function is decreasing.Hmm, so between t = -1 and t = 0, the function changes from decreasing to increasing. Maybe the minimum occurs somewhere in between.Let me try t = -0.5:f(-0.5) = sqrt[(2*(0.25) - 3)² + (4*(-0.5) + 1)²] + 2*(0.25 + 1) = sqrt[(0.5 - 3)² + (-2 + 1)²] + 2*(1.25) = sqrt[(-2.5)² + (-1)²] + 2.5 = sqrt[6.25 + 1] + 2.5 = sqrt(7.25) + 2.5 ≈ 2.692 + 2.5 ≈ 5.192f'(-0.5) = (8*(-0.5)^3 + 4*(-0.5) + 4)/sqrt[(-2.5)^2 + (-1)^2] + 4*(-0.5) = (8*(-0.125) - 2 + 4)/sqrt[6.25 + 1] - 2 = (-1 - 2 + 4)/sqrt(7.25) - 2 = (1)/2.692 - 2 ≈ 0.371 - 2 ≈ -1.629 < 0Still decreasing.Try t = -0.25:f(-0.25) = sqrt[(2*(0.0625) - 3)² + (4*(-0.25) + 1)²] + 2*(0.0625 + 1) = sqrt[(0.125 - 3)² + (-1 + 1)²] + 2*(1.0625) = sqrt[(-2.875)^2 + 0] + 2.125 ≈ sqrt(8.266) + 2.125 ≈ 2.875 + 2.125 = 5f'(-0.25) = (8*(-0.25)^3 + 4*(-0.25) + 4)/sqrt[(-2.875)^2 + 0] + 4*(-0.25) = (8*(-0.015625) - 1 + 4)/2.875 - 1 = (-0.125 - 1 + 4)/2.875 - 1 ≈ (2.875)/2.875 - 1 = 1 - 1 = 0Wow, so at t = -0.25, the derivative is zero. So, this might be a critical point.Let me check f(-0.25):f(-0.25) = sqrt[(2*(0.0625) - 3)^2 + (4*(-0.25) + 1)^2] + 2*(0.0625 + 1) = sqrt[(0.125 - 3)^2 + (-1 + 1)^2] + 2*(1.0625) = sqrt[(-2.875)^2 + 0] + 2.125 ≈ 2.875 + 2.125 = 5So, f(-0.25) = 5.Now, let's check the value of |AM| + |MF|.Wait, earlier I had |AM| + |MF| ≥ |MC| - 1 + |MF|, and |MC| + |MF| was minimized at 5, so |AM| + |MF| ≥ 5 - 1 = 4.But wait, at t = -0.25, f(t) = |MC| + |MF| = 5, so |AM| + |MF| ≥ 5 - 1 = 4.Is this the minimal value? Let me check.If |MC| + |MF| is minimized at 5, then |AM| + |MF| is minimized at 5 - 1 = 4.But I need to confirm if this is achievable.So, when |MC| + |MF| is minimized at 5, then |AM| + |MF| is at least 4. But is there a point A on the circle such that |AM| = |MC| - 1?Yes, because the minimal |AM| is |MC| - 1, which occurs when A is on the line segment MC, extended from C towards M.So, when M is at t = -0.25, which is (2*(-0.25)^2, 4*(-0.25)) = (2*(0.0625), -1) = (0.125, -1).Wait, that's interesting. So, M is at (0.125, -1). Point C is at (3, -1). So, the line segment MC is horizontal, from (3, -1) to (0.125, -1). The distance |MC| is 3 - 0.125 = 2.875, which matches our earlier calculation.So, the minimal |AM| is |MC| - 1 = 2.875 - 1 = 1.875.But wait, |AM| + |MF| = 1.875 + |MF|.But |MF| is 2(t² + 1) = 2*(0.0625 + 1) = 2*(1.0625) = 2.125.So, |AM| + |MF| = 1.875 + 2.125 = 4.Therefore, the minimal value is indeed 4.So, the minimal value of |AM| + |MF| is 4.

💡Okay, so I have this problem here: If a number is divided by 5, the remainder is 3. What is the remainder when four times the number plus 6 is divided by 5? Hmm, let me try to figure this out step by step.First, I need to understand what it means when a number is divided by 5 and leaves a remainder of 3. That means if I take the number and divide it by 5, it doesn't divide evenly; instead, there's a leftover of 3. So, mathematically, I can represent this number as 5k + 3, where k is some integer. For example, if k is 1, the number would be 8 because 5*1 + 3 = 8. If k is 2, the number would be 13, and so on.Now, the problem is asking about four times this number plus 6, and then what the remainder is when that result is divided by 5. So, I need to compute 4*(number) + 6 and then find the remainder when this new number is divided by 5.Let me write this out using the expression I have for the number. If the number is 5k + 3, then four times the number would be 4*(5k + 3). Let me calculate that:4*(5k + 3) = 4*5k + 4*3 = 20k + 12.Now, I need to add 6 to this result:20k + 12 + 6 = 20k + 18.So, the expression we're interested in is 20k + 18. Now, we need to find the remainder when this is divided by 5.To find the remainder when 20k + 18 is divided by 5, I can look at each term separately. First, 20k divided by 5 is straightforward because 20 is a multiple of 5. Specifically, 20 divided by 5 is 4, so 20k divided by 5 is 4k, which is an integer with no remainder. Therefore, the remainder comes solely from the second term, which is 18.Now, I need to find the remainder when 18 is divided by 5. Let's do that division: 5 goes into 18 three times (because 5*3=15), and the remainder is 18 - 15 = 3. So, 18 divided by 5 leaves a remainder of 3.Putting it all together, since 20k divided by 5 leaves no remainder and 18 divided by 5 leaves a remainder of 3, the entire expression 20k + 18 divided by 5 will also leave a remainder of 3.Wait, let me double-check that to make sure I didn't make a mistake. If I take a specific example, say k=1, then the original number is 8. Four times 8 is 32, plus 6 is 38. Dividing 38 by 5 gives 7 with a remainder of 3. Yep, that matches what I found earlier.Another example: k=2, the number is 13. Four times 13 is 52, plus 6 is 58. Dividing 58 by 5 gives 11 with a remainder of 3. Again, the remainder is 3. Okay, so it seems consistent.I think I'm confident now that the remainder is indeed 3 when four times the number plus 6 is divided by 5.