answer：Okay, so I have this problem here: "The distance from a point A on the parabola y² = 4x to its focus F is 4. What is the slope of line AF?" Hmm, let me try to figure this out step by step.First, I remember that for a parabola in the form y² = 4ax, the focus is at (a, 0). In this case, comparing y² = 4x to y² = 4ax, I can see that 4a = 4, so a = 1. That means the focus F is at (1, 0). Got that down.Now, the problem says that the distance from point A on the parabola to the focus F is 4. So, point A is somewhere on the parabola, and the straight-line distance from A to F is 4 units. I need to find the slope of the line connecting A and F.Let me denote point A as (x, y). Since A is on the parabola y² = 4x, it must satisfy this equation. So, y² = 4x. That's one equation.Next, the distance between A(x, y) and F(1, 0) is given as 4. The distance formula between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)² + (y2 - y1)²]. Applying that here, the distance AF is sqrt[(x - 1)² + (y - 0)²] = sqrt[(x - 1)² + y²] = 4.So, I can write the equation:sqrt[(x - 1)² + y²] = 4If I square both sides to eliminate the square root, I get:(x - 1)² + y² = 16Now, I have two equations:1. y² = 4x (from the parabola)2. (x - 1)² + y² = 16 (from the distance)I can substitute equation 1 into equation 2 to solve for x. Let's do that.Substitute y² = 4x into the second equation:(x - 1)² + 4x = 16Let me expand (x - 1)²:(x² - 2x + 1) + 4x = 16Combine like terms:x² - 2x + 1 + 4x = 16Simplify:x² + 2x + 1 = 16Wait, that simplifies to:x² + 2x + 1 = 16Hmm, x² + 2x + 1 is a perfect square, right? That's (x + 1)².So, (x + 1)² = 16Taking square roots on both sides:x + 1 = ±4So, x = -1 ± 4Therefore, x can be:x = -1 + 4 = 3orx = -1 - 4 = -5Wait, hold on. The parabola y² = 4x opens to the right, so x must be greater than or equal to 0. Because if x were negative, y² would be negative, which isn't possible. So, x = -5 is not a valid solution here.Therefore, the only valid x-coordinate is x = 3.Now, plugging x = 3 back into the equation y² = 4x:y² = 4 * 3 = 12So, y = ±√12 = ±2√3Therefore, point A has coordinates (3, 2√3) or (3, -2√3).Now, I need to find the slope of line AF. Since F is at (1, 0), let's compute the slope between (1, 0) and (3, 2√3), and also between (1, 0) and (3, -2√3).The slope formula is (y2 - y1)/(x2 - x1).First, for point A(3, 2√3):Slope = (2√3 - 0)/(3 - 1) = (2√3)/2 = √3Second, for point A(3, -2√3):Slope = (-2√3 - 0)/(3 - 1) = (-2√3)/2 = -√3So, the slope of line AF is either √3 or -√3, depending on whether point A is above or below the x-axis.Let me just double-check my steps to make sure I didn't make any mistakes.1. Identified the focus correctly as (1, 0). That seems right because for y² = 4ax, focus is at (a, 0), which is (1, 0) here.2. Set up the distance formula correctly between A(x, y) and F(1, 0). Squared both sides to get rid of the square root, which is standard.3. Substituted y² = 4x into the distance equation, expanded correctly, and simplified. Got to x² + 2x + 1 = 16, recognized it as (x + 1)² = 16, solved for x, got x = 3 and x = -5, dismissed x = -5 because it's not on the parabola. That seems solid.4. Plugged x = 3 back into y² = 4x to get y = ±2√3. Correct.5. Calculated the slope for both points, got √3 and -√3. That makes sense because the two points are symmetric with respect to the x-axis, so their slopes are negatives of each other.So, I think I did everything correctly. The slope is either √3 or -√3.**Final Answer**The slope of line AF is boxed{pm sqrt{3}}.

answer：First, I need to determine the percentage of apples that are mixed with other products. Initially, it's 30% for the first 8 million tons. Since the production is exactly 8 million tons, there are no additional million tons to consider, so the percentage remains at 30%.Next, I calculate the amount of apples mixed with other products by finding 30% of 8 million tons:0.30 × 8 = 2.4 million tons.Then, I subtract this amount from the total production to find the remaining apples:8 million tons - 2.4 million tons = 5.6 million tons.Finally, I determine the amount of apples sold fresh by calculating 40% of the remaining apples:0.40 × 5.6 = 2.24 million tons.

answer：To solve the equation (frac{c - 37}{3} = frac{3c + 7}{8}), follow these steps:1. **Eliminate the denominators** by multiplying both sides by the least common denominator (LCD) of 3 and 8, which is 24: [ 24 times frac{c - 37}{3} = 24 times frac{3c + 7}{8} ] Simplifying both sides: [ 8(c - 37) = 3(3c + 7) ]2. **Expand both sides** of the equation: [ 8c - 296 = 9c + 21 ]3. **Rearrange the equation** to isolate (c): [ 8c - 9c = 21 + 296 ] [ -c = 317 ] [ c = -317 ]Thus, the solution is (c = -317).

answer：Okay, so I have this geometry problem here. It says: Given a triangle XYZ where a circle with center D is inscribed, find angle YDZ if angle XYZ is 75 degrees and angle XZY is 53 degrees. Hmm, let me try to visualize this.First, I know that an inscribed circle, or incircle, touches all three sides of the triangle. The center of this circle is called the incenter, which is the point where the angle bisectors of the triangle meet. So, point D is the incenter of triangle XYZ.Now, the problem is asking for angle YDZ. That means I need to find the measure of the angle formed at point D by points Y and Z. To do this, I think I need to use some properties of the incenter and angle bisectors.Let me recall that the incenter is equidistant from all sides of the triangle, and it's the intersection point of the angle bisectors. So, each angle bisector will split the corresponding angle into two equal parts.Given that angle XYZ is 75 degrees and angle XZY is 53 degrees, I can find the remaining angle at point X. Since the sum of the angles in a triangle is always 180 degrees, I can calculate angle YXZ as follows:Angle YXZ = 180° - angle XYZ - angle XZYAngle YXZ = 180° - 75° - 53°Angle YXZ = 180° - 128°Angle YXZ = 52°Okay, so angle at X is 52 degrees. Now, since D is the incenter, the lines from D to the vertices (XD, YD, ZD) are the angle bisectors. That means each of these lines splits the respective angles into two equal parts.So, angle YDZ is formed by the intersection of the bisectors of angles at Y and Z. Wait, actually, angle YDZ is the angle at D between points Y and Z. Hmm, maybe I need to think about the angles around point D.Let me try to sketch this in my mind. Triangle XYZ with incenter D. So, from D, there are three lines: one to X, one to Y, and one to Z. Each of these lines bisects the angles at those vertices.So, angle YDZ is the angle at D between the bisectors of angles Y and Z. To find this angle, I might need to use the fact that the sum of the angles around point D is 360 degrees. But I'm not sure if that's the right approach.Alternatively, maybe I can use the formula for the angle between two angle bisectors. I remember that in a triangle, the angle between two angle bisectors can be found using the formula:Angle between bisectors = 90° + (angle at the vertex)/2Wait, no, that's for the angle between an angle bisector and a median or something else. Maybe I'm mixing up formulas.Let me think differently. Since D is the incenter, the angles at D related to the triangle's angles can be found using the fact that each angle bisector divides the angle into two equal parts.So, angle YDZ is the angle at D between the bisectors of angles Y and Z. Let me denote the bisected angles.Angle at Y is 75°, so each bisected angle is 75°/2 = 37.5°Angle at Z is 53°, so each bisected angle is 53°/2 = 26.5°Now, in triangle YDZ, we have two angles at Y and Z, which are 37.5° and 26.5° respectively. Wait, no, triangle YDZ is not the same as triangle XYZ. Hmm, maybe I need to consider triangle YDZ separately.Actually, triangle YDZ is a smaller triangle within triangle XYZ, with vertices at Y, D, and Z. To find angle YDZ, I can use the fact that the sum of the angles in triangle YDZ is 180°, so if I can find the other two angles at Y and Z in triangle YDZ, I can subtract their sum from 180° to get angle YDZ.But wait, in triangle YDZ, the angles at Y and Z are not the same as the bisected angles from triangle XYZ. Because in triangle YDZ, the angles at Y and Z are actually the angles between the sides of the triangle and the bisectors.Hmm, maybe I need to use the formula for the incenter angles. I think there's a formula that relates the angles at the incenter to the angles of the original triangle.I recall that the angle at the incenter opposite to a vertex is equal to 90° plus half the angle at that vertex. Wait, let me check that.Actually, the angle between two angle bisectors can be calculated using the formula:Angle between bisectors = 90° + (angle at the opposite vertex)/2So, in this case, angle YDZ would be equal to 90° plus half of angle YXZ.Wait, angle YXZ is 52°, so half of that is 26°. Therefore, angle YDZ would be 90° + 26° = 116°. Hmm, but that doesn't seem right because 116° seems too large for an angle inside the triangle.Wait, maybe I got the formula wrong. Let me think again. The formula might actually be:Angle between two angle bisectors = 90° + (angle at the opposite vertex)/2But in this case, angle YDZ is the angle between the bisectors of angles Y and Z, so the opposite vertex would be X. Therefore, angle YDZ = 90° + (angle YXZ)/2.Since angle YXZ is 52°, half of that is 26°, so angle YDZ = 90° + 26° = 116°. But wait, if I consider the incenter, the angles around D should be less than 180°, but 116° is possible.Wait, but let me verify this with another approach. Maybe using coordinates or trigonometry.Alternatively, I can use the fact that in triangle XYZ, the incenter D creates three smaller triangles: XYD, YZD, and ZXD. Each of these triangles has angles that are related to the original angles of triangle XYZ.In triangle YDZ, the angles at Y and Z are half of the original angles at Y and Z because D is the incenter. So, angle at Y in triangle YDZ is 75°/2 = 37.5°, and angle at Z is 53°/2 = 26.5°. Therefore, the sum of these two angles is 37.5° + 26.5° = 64°. Therefore, angle YDZ = 180° - 64° = 116°.Wait, that matches the previous result. So, angle YDZ is 116°. But earlier, I thought it might be 26°, which was incorrect. So, I need to figure out why I initially thought 26°, but that was wrong.Wait, maybe I confused angle YDZ with another angle. Let me clarify.In triangle XYZ, D is the incenter. So, the lines from D to the vertices are the angle bisectors. Therefore, angle YDZ is the angle at D between the bisectors of angles Y and Z.To find this angle, I can use the formula:Angle YDZ = 90° + (angle YXZ)/2Since angle YXZ is 52°, then angle YDZ = 90° + 26° = 116°.Alternatively, using the triangle angle sum approach in triangle YDZ:Angles at Y and Z in triangle YDZ are half of the original angles at Y and Z, so 37.5° and 26.5°, summing to 64°, so angle YDZ = 180° - 64° = 116°.Therefore, angle YDZ is 116°.Wait, but in the initial problem, the user provided a solution that concluded angle YDZ is 26°, which contradicts my current conclusion. So, I must have made a mistake somewhere.Let me go back to the problem statement. It says: "find angle YDZ if angle XYZ = 75° and angle XZY = 53°." So, angle at Y is 75°, angle at Z is 53°, so angle at X is 52°, as calculated before.Now, the incenter D is where the angle bisectors meet. So, the bisector of angle Y (75°) will split it into two angles of 37.5° each, and the bisector of angle Z (53°) will split it into two angles of 26.5° each.Now, considering triangle YDZ, which has vertices at Y, D, and Z. The angles at Y and Z in this triangle are 37.5° and 26.5°, respectively. Therefore, the third angle at D, which is angle YDZ, is 180° - 37.5° - 26.5° = 116°.So, angle YDZ is 116°, not 26°. Therefore, the initial solution provided by the user was incorrect.Wait, but maybe I'm misunderstanding the notation. Maybe angle YDZ is not the angle at D in triangle YDZ, but rather the angle between DY and DZ as seen from point Y or Z. No, that doesn't make sense. Angle YDZ should be the angle at D between points Y and Z.Alternatively, perhaps the user made a mistake in their initial solution by considering angle YDZ as half of angle YXZ, which is 26°, but that's incorrect because angle YDZ is not half of angle YXZ, but rather related to the sum of half-angles from Y and Z.Wait, let me think again. Maybe I can use the formula for the angle between two angle bisectors. The formula is:Angle between two angle bisectors = 90° + (angle at the opposite vertex)/2In this case, the angle at the opposite vertex is angle YXZ, which is 52°, so the angle between the bisectors of angles Y and Z is 90° + 26° = 116°, which confirms my previous result.Therefore, the correct measure of angle YDZ is 116°, not 26°. So, the initial solution provided was incorrect.Wait, but maybe the user was considering a different angle, such as angle ZDY or something else. Let me double-check the notation.In angle YDZ, the vertex is D, with segments DY and DZ. So, it's the angle at D between Y and Z, which is indeed 116°.Therefore, the correct answer should be 116°, not 26°. So, the initial solution was wrong, and the correct measure is 116°.

answer：First, I need to determine the total number of people across all three groups. Group A has 18 people, Group B has 24 people, and Group C has 30 people. Adding these together gives a total of 72 people.Next, I'll calculate the proportion of Group B within the entire population. Since Group B has 24 people out of 72, the proportion is 24 divided by 72, which simplifies to 1/3.Now, I need to apply this proportion to the total sample size of 12 people. Multiplying 12 by 1/3 gives 4. Therefore, 4 people should be drawn from Group B to maintain the same proportion in the sample as in the population.This method ensures that the sample accurately represents the composition of the entire population across all three groups.

answer：First, I need to simplify the expression (frac{180}{16} cdot frac{5}{120} cdot frac{8}{3}). I'll start by identifying common factors between the numerators and denominators to make the fractions easier to work with. Looking at the first fraction, (frac{180}{16}), I notice that both 180 and 16 can be divided by 4. Dividing 180 by 4 gives 45, and dividing 16 by 4 gives 4. So, (frac{180}{16}) simplifies to (frac{45}{4}).Next, I'll simplify (frac{5}{120}). Both 5 and 120 can be divided by 5. Dividing 5 by 5 gives 1, and dividing 120 by 5 gives 24. Therefore, (frac{5}{120}) simplifies to (frac{1}{24}).Now, the expression looks like (frac{45}{4} cdot frac{1}{24} cdot frac{8}{3}).I'll multiply the numerators together and the denominators together: (frac{45 cdot 1 cdot 8}{4 cdot 24 cdot 3}).Calculating the numerator: (45 times 1 = 45), and (45 times 8 = 360).Calculating the denominator: (4 times 24 = 96), and (96 times 3 = 288).So, the expression simplifies to (frac{360}{288}).To reduce this fraction, I'll divide both the numerator and the denominator by their greatest common divisor, which is 72. Dividing 360 by 72 gives 5, and dividing 288 by 72 gives 4.Therefore, the simplified form of the expression is (frac{5}{4}).