answer：First, I need to determine the normal price of each 4 oz package of butter. Let's denote this normal price as ( P ).The woman has two options to purchase 16 oz of butter:1. Buy a single 16 oz package for 7.2. Buy an 8 oz package for 4 and use a coupon to get two 4 oz packages at a 50% discount.Since the lowest price she can pay is 6, we focus on the second option.The total amount of butter needed is 16 oz. She already has an 8 oz package, so she needs an additional 8 oz, which she can get by purchasing two 4 oz packages.With the coupon, each 4 oz package is discounted by 50%, so the price for each discounted 4 oz package is ( 0.5P ).The total cost for the two discounted 4 oz packages is ( 2 times 0.5P = P ).Adding the cost of the 8 oz package, the total cost is:[4 + P = 6]Solving for ( P ):[P = 6 - 4 = 2]Therefore, the normal price of each 4 oz package is 2.

answer：Okay, so I have this problem here with two vectors, a and b. Vector a is (1, m) and vector b is (3, -2). The problem says that the sum of vectors a and b is perpendicular to vector b, and I need to find the value of m. Hmm, okay, let me think about how to approach this.First, I remember that if two vectors are perpendicular, their dot product is zero. So, if (a + b) is perpendicular to b, then the dot product of (a + b) and b should be zero. That makes sense. So, my first step should be to find the vector a + b.Let me write that down. Vector a is (1, m) and vector b is (3, -2). So, adding these two vectors component-wise, I get:a + b = (1 + 3, m + (-2)) = (4, m - 2)Okay, so the sum of vectors a and b is (4, m - 2). Now, I need to find the dot product of this resulting vector with vector b and set it equal to zero because they are perpendicular.The dot product of two vectors (x1, y1) and (x2, y2) is x1*x2 + y1*y2. So, applying that to (4, m - 2) and (3, -2):Dot product = 4*3 + (m - 2)*(-2)Let me compute that:4*3 is 12, and (m - 2)*(-2) is -2m + 4. So, adding those together:12 + (-2m + 4) = 12 - 2m + 4Wait, that simplifies to 16 - 2m. Since the dot product is zero, I can set up the equation:16 - 2m = 0Now, solving for m:16 - 2m = 0 -2m = -16 m = (-16)/(-2) m = 8So, m should be 8. Let me double-check my steps to make sure I didn't make any mistakes.1. I added vectors a and b correctly: (1+3, m + (-2)) = (4, m - 2). 2. I calculated the dot product correctly: 4*3 + (m - 2)*(-2) = 12 - 2m + 4 = 16 - 2m. 3. I set the dot product equal to zero because the vectors are perpendicular: 16 - 2m = 0. 4. Solved for m: m = 8.Everything seems to check out. I don't see any errors in my calculations or reasoning. So, I'm confident that m is indeed 8.**Final Answer**The value of ( m ) is boxed{8}.

answer：Alright, so I have this problem here about a normal distribution. Let me try to understand it step by step. The random variable ξ follows a normal distribution N(40, σ²). That means it has a mean of 40 and a variance of σ². I don't know the value of σ, but maybe I don't need it to solve the problem.The problem says that P(ξ < 30) = 0.2. So, the probability that ξ is less than 30 is 20%. I need to find P(30 < ξ < 50). Hmm, okay. So, that's the probability that ξ is between 30 and 50.Since it's a normal distribution, I know the total probability is 1, or 100%. So, if I can figure out the probabilities on either side of 30 and 50, I can find the probability between them.First, let me recall that in a normal distribution, the probabilities are symmetric around the mean. The mean here is 40. So, 30 is 10 units below the mean, and 50 is 10 units above the mean. That seems symmetrical. So, maybe the probability above 50 is the same as the probability below 30?Wait, the problem says P(ξ < 30) = 0.2. So, if 30 is 10 below the mean, and 50 is 10 above, then P(ξ > 50) should also be 0.2, right? Because of the symmetry.So, if P(ξ < 30) = 0.2 and P(ξ > 50) = 0.2, then the total probability outside the range 30 to 50 is 0.2 + 0.2 = 0.4. Therefore, the probability inside the range 30 to 50 should be 1 - 0.4 = 0.6.Wait, let me check that again. If P(ξ < 30) is 0.2, and because of symmetry, P(ξ > 50) is also 0.2, then the total probability outside 30 and 50 is 0.4. So, the probability between 30 and 50 is 1 - 0.4 = 0.6. That makes sense.But just to be thorough, maybe I should standardize the variable and use Z-scores to confirm. Let me try that.The Z-score formula is Z = (X - μ) / σ. Here, μ is 40. So, for X = 30, Z = (30 - 40)/σ = (-10)/σ. Similarly, for X = 50, Z = (50 - 40)/σ = 10/σ.Given that P(ξ < 30) = 0.2, which is the same as P(Z < (-10)/σ) = 0.2. From standard normal distribution tables, the Z-score corresponding to 0.2 probability is approximately -0.84. So, (-10)/σ = -0.84. Solving for σ, we get σ = 10 / 0.84 ≈ 11.9048.Now, let's find P(30 < ξ < 50). That's the same as P(-0.84 < Z < 0.84). From the standard normal table, P(Z < 0.84) is about 0.7995, and P(Z < -0.84) is about 0.2005. So, the probability between -0.84 and 0.84 is 0.7995 - 0.2005 = 0.599, which is approximately 0.6.Okay, so both methods give me the same result. That makes me more confident that the answer is 0.6.So, the probability that ξ is between 30 and 50 is 0.6, which corresponds to option C.**Final Answer**boxed{C}

answer：Alright, so I have this geometry problem here with a diagram involving several right-angled triangles. Let me try to visualize and understand the setup first. There are triangles ABF, BCF, and CDF, all right-angled. The angles at ABF and BCF are both 60 degrees, and the angle at CDF is 45 degrees. The length of AF is given as 48 units. I need to find the length of CF.Okay, let me break this down step by step. First, I should probably sketch a rough diagram based on the Asymptote code provided. It shows points A, B, C, D, and F connected in a certain way. A is connected to B, which is connected to C, then to D, then to F, and back to A. Also, there are lines from B to F and from C to F. So, it's a bit of a complex figure with multiple triangles sharing sides.Starting with triangle ABF. It's a right-angled triangle with angle ABF being 60 degrees. Since it's right-angled, one of the angles is 90 degrees. But the given angle is 60 degrees, so the other non-right angle must be 30 degrees because the angles in a triangle add up to 180 degrees. So, triangle ABF is a 30-60-90 triangle.In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where 1 is the side opposite the 30-degree angle, √3 is opposite the 60-degree angle, and 2 is the hypotenuse. Given that AF is 48, which is the hypotenuse of triangle ABF, I can find the lengths of the other sides.Let me denote the sides:- AF (hypotenuse) = 48- AB (opposite 60 degrees) = (48) * (√3 / 2) = 24√3- BF (opposite 30 degrees) = (48) / 2 = 24So, BF is 24 units long.Moving on to triangle BCF. It's also a right-angled triangle with angle BCF being 60 degrees. Again, this is a 30-60-90 triangle. The hypotenuse here is BF, which we found to be 24. So, let's find the sides of triangle BCF.- BF (hypotenuse) = 24- BC (opposite 60 degrees) = (24) * (√3 / 2) = 12√3- CF (opposite 30 degrees) = (24) / 2 = 12So, CF is 12 units long. Wait, but before I conclude, I should check if triangle CDF affects this. Triangle CDF is right-angled with angle CDF being 45 degrees, making it a 45-45-90 triangle. In such triangles, both legs are equal, and the hypotenuse is leg * √2.But since we're dealing with CF, which is a side in triangle BCF, and triangle CDF is connected to CF, I need to ensure that CF is consistent in both triangles.In triangle CDF, CF is one of the legs, and since angle CDF is 45 degrees, both legs should be equal. So, if CF is 12, then DF should also be 12. Then, the hypotenuse CD would be 12√2. But wait, in triangle BCF, BC is 12√3, and CD is another segment. I need to make sure that the lengths are consistent throughout the figure.Looking back, triangle ABF has AF = 48, AB = 24√3, BF = 24. Triangle BCF has BF = 24, BC = 12√3, CF = 12. Triangle CDF has CF = 12, DF = 12, and CD = 12√2. Is there any inconsistency here? Let me check the coordinates given in the Asymptote code. Point F is at (24, 0), which makes sense because BF is 24 units along the x-axis. Point B is at (0, 0), so BF is along the x-axis from (0,0) to (24,0). Point A is at (0, 41.57), which is approximately (0, 24√3) since 24√3 ≈ 41.57. That checks out.Point C is at (18, -10.392). Let me see, 10.392 is approximately 6√3, which is about 10.392. So, point C is 18 units along the x-axis and -6√3 along the y-axis. That would make BC the distance from B(0,0) to C(18, -6√3). Let's calculate that distance:Distance BC = √[(18 - 0)^2 + (-6√3 - 0)^2] = √[324 + 108] = √[432] = √[16*27] = 4*3√3 = 12√3. That matches our earlier calculation.Now, point D is at (27, -5.196). 5.196 is approximately 3√3, which is about 5.196. So, point D is 27 units along the x-axis and -3√3 along the y-axis. Let's find the distance from C(18, -6√3) to D(27, -3√3):Distance CD = √[(27 - 18)^2 + (-3√3 - (-6√3))^2] = √[81 + (3√3)^2] = √[81 + 27] = √108 = √[36*3] = 6√3. Wait, but earlier I thought CD should be 12√2. Hmm, that's conflicting.Wait, maybe I made a mistake in interpreting triangle CDF. Let me see. Triangle CDF is right-angled at D, with angle CDF being 45 degrees. So, the right angle is at D, and angle at D is 45 degrees, meaning the other angle at F is also 45 degrees. Therefore, triangle CDF is a 45-45-90 triangle, so CF should be equal to DF, and CD should be CF * √2.But according to the coordinates, CF is from C(18, -6√3) to F(24, 0). Let's calculate CF:Distance CF = √[(24 - 18)^2 + (0 - (-6√3))^2] = √[36 + 108] = √144 = 12. So, CF is indeed 12 units.Then, DF is from D(27, -3√3) to F(24, 0):Distance DF = √[(24 - 27)^2 + (0 - (-3√3))^2] = √[9 + 27] = √36 = 6. Wait, that's only 6 units, but earlier I thought DF should be equal to CF, which is 12. That's a discrepancy.Hold on, maybe I misapplied the triangle CDF. If triangle CDF is right-angled at D with angle CDF = 45 degrees, then sides CD and DF should be equal because it's a 45-45-90 triangle. But according to the coordinates, CD is 6√3 and DF is 6, which are not equal. That suggests an inconsistency.Wait, perhaps the right angle is not at D but somewhere else? Let me check the Asymptote code again. It says triangle CDF is right-angled, with angle CDF = 45 degrees. So, the right angle must be at F or at C. But angle at D is 45 degrees, so the right angle can't be at D because that would make angle D 90 degrees, conflicting with the given 45 degrees.Therefore, the right angle must be at C or at F. If it's at C, then angle at C is 90 degrees, and angle at D is 45 degrees, making angle at F also 45 degrees. So, triangle CDF would have sides CD and CF equal, with DF as the hypotenuse. But according to the coordinates, CF is 12, DF is 6, which doesn't fit.Alternatively, if the right angle is at F, then angle at F is 90 degrees, angle at D is 45 degrees, making angle at C also 45 degrees. So, sides CF and DF would be equal, and CD would be CF * √2. But from coordinates, CF is 12, DF is 6, which again doesn't fit.Hmm, this is confusing. Maybe I need to recast my approach. Let's try to use coordinates to find CF.From the Asymptote code, point F is at (24, 0). Point C is at (18, -10.392). So, the distance CF is √[(24 - 18)^2 + (0 - (-10.392))^2] = √[36 + 108] = √144 = 12. So, CF is indeed 12 units.But earlier, when I tried to compute DF, it came out as 6 units, which doesn't align with triangle CDF being a 45-45-90 triangle. Maybe the issue is that I'm misinterpreting the right angle's position. Let me check the Asymptote code again.The Asymptote code draws A-B-C-D-F-A, and also draws B-F and C-F. So, triangle CDF is formed by points C, D, F. Since it's right-angled, and angle at D is 45 degrees, the right angle must be at F or at C.If the right angle is at F, then CF and DF are the legs, and CD is the hypotenuse. Given that angle at D is 45 degrees, both legs should be equal, meaning CF = DF. But from coordinates, CF is 12 and DF is 6, which contradicts.If the right angle is at C, then CD and CF are the legs, and DF is the hypotenuse. Given angle at D is 45 degrees, the legs should be equal, so CD = CF. But CD is 6√3 ≈ 10.392, and CF is 12, which again contradicts.Wait, maybe the right angle is at D? But angle at D is given as 45 degrees, so it can't be the right angle. Therefore, perhaps the Asymptote code is misleading, or I'm misinterpreting the angles.Alternatively, maybe triangle CDF is right-angled at C, with angle at D being 45 degrees. So, angle at C is 90 degrees, angle at D is 45 degrees, making angle at F also 45 degrees. Therefore, sides CD and CF should be equal, but from coordinates, CD is 6√3 and CF is 12, which are not equal. So, that doesn't work.Wait, perhaps I made a mistake in calculating CD. Let me recalculate CD using the coordinates. Point C is at (18, -10.392), which is (18, -6√3). Point D is at (27, -5.196), which is (27, -3√3). So, the distance CD is √[(27-18)^2 + (-3√3 - (-6√3))^2] = √[81 + (3√3)^2] = √[81 + 27] = √108 = 6√3. So, CD is 6√3, which is approximately 10.392.But CF is 12, so if triangle CDF is right-angled at C, then CD and CF would be legs, and DF would be the hypotenuse. So, DF should be √[(6√3)^2 + 12^2] = √[108 + 144] = √252 = 6√7 ≈ 15.874. But from coordinates, DF is only 6 units. That's a big discrepancy.Alternatively, if triangle CDF is right-angled at F, then CF and DF are legs, and CD is the hypotenuse. So, CD should be √[12^2 + 6^2] = √[144 + 36] = √180 = 6√5 ≈ 13.416. But from coordinates, CD is 6√3 ≈ 10.392, which doesn't match.This suggests that either the coordinates are incorrect, or my interpretation of the triangles is wrong. But the problem states that triangles ABF, BCF, and CDF are right-angled, with angles at ABF, BCF, and CDF being 60°, 60°, and 45° respectively.Wait, maybe the right angles are not at the points I thought. For triangle ABF, it's right-angled, with angle at B being 60°, so the right angle must be at F or A. Similarly for the others.Let me clarify:- Triangle ABF: right-angled, angle at B is 60°, so the right angle must be at A or F.But since AF is the hypotenuse, which is given as 48, and angle at B is 60°, then the right angle must be at A or F. If angle at B is 60°, and it's a right-angled triangle, then the right angle must be at A or F.Wait, in a triangle, the right angle is the largest angle, so if angle at B is 60°, the right angle must be at A or F. Let's assume it's at A. Then, angle at A is 90°, angle at B is 60°, so angle at F is 30°. Then, sides:- AF (hypotenuse) = 48- AB (opposite 60°) = 24√3- BF (opposite 30°) = 24That matches our earlier calculation.Similarly, triangle BCF is right-angled, angle at B is 60°, so the right angle must be at C or F. If right angle is at C, then angle at B is 60°, angle at F is 30°, making sides:- BF (hypotenuse) = 24- BC (opposite 60°) = 12√3- CF (opposite 30°) = 12That's consistent.Now, triangle CDF is right-angled, angle at D is 45°, so the right angle must be at C or F. If right angle is at C, then angle at D is 45°, making angle at F also 45°, so sides CD and CF should be equal. But from coordinates, CF is 12, CD is 6√3 ≈ 10.392, which isn't equal.Alternatively, if the right angle is at F, then angle at D is 45°, making angle at C also 45°, so sides CF and DF should be equal. From coordinates, CF is 12, DF is 6, which isn't equal.This inconsistency suggests that perhaps the right angle in triangle CDF is not at C or F, but at D. But angle at D is given as 45°, so it can't be the right angle. Therefore, there must be a misunderstanding in the problem setup.Wait, maybe the right angle in triangle CDF is at C, but angle at D is 45°, so it's a 45-45-90 triangle. But then CD and CF should be equal, but they aren't. Alternatively, maybe the right angle is at F, making CF and DF equal, but they aren't.This is perplexing. Perhaps I need to approach this differently, using coordinate geometry.Given the coordinates:- A(0, 41.57) ≈ (0, 24√3)- B(0, 0)- C(18, -10.392) ≈ (18, -6√3)- D(27, -5.196) ≈ (27, -3√3)- F(24, 0)Let me verify the triangles:1. Triangle ABF: Points A(0,24√3), B(0,0), F(24,0). It's a right triangle with right angle at B? Wait, no, because AB is vertical, BF is horizontal, so the right angle is at B. But angle at B is given as 60°, which contradicts. Wait, no, angle ABF is 60°, which is at B. So, triangle ABF has angle at B of 60°, and it's right-angled. So, the right angle must be at A or F.But in coordinates, AB is vertical, BF is horizontal, so the angle at B is 90°, but the problem states angle ABF is 60°. That's a contradiction. Therefore, the Asymptote code might not be to scale, or the coordinates are approximate.Given that, perhaps I should ignore the coordinates and focus on the given angles and lengths.So, starting fresh:- Triangle ABF: right-angled, angle at B is 60°, AF = 48.Since it's right-angled and has a 60° angle, it must be a 30-60-90 triangle. The side opposite 30° is half the hypotenuse, so BF = 24. The side opposite 60° is 24√3, so AB = 24√3.- Triangle BCF: right-angled, angle at B is 60°, hypotenuse BF = 24.Again, 30-60-90 triangle. The side opposite 30° is 12, so CF = 12. The side opposite 60° is 12√3, so BC = 12√3.- Triangle CDF: right-angled, angle at D is 45°, so it's a 45-45-90 triangle. Therefore, legs CF and DF are equal, and hypotenuse CD is CF√2.But from triangle BCF, CF = 12, so DF should also be 12, and CD should be 12√2.However, in the coordinate system, DF is only 6 units. This suggests that either the coordinates are misleading, or there's a scaling factor I'm missing.Wait, perhaps the Asymptote code is just a rough sketch, and the coordinates are approximate. The exact coordinates might not align perfectly with the calculated lengths, especially since AF is given as 48, but in the Asymptote code, AF is from (0,41.57) to (24,0), which is approximately 48 units (since √(24² + 41.57²) ≈ √(576 + 1729) ≈ √2305 ≈ 48). So, it's consistent in that sense.But the issue with triangle CDF remains. If CF is 12, then DF should also be 12, but in the coordinates, DF is only 6. Maybe the Asymptote code is not to scale, or perhaps I'm misinterpreting the triangles.Alternatively, perhaps triangle CDF is not connected as I thought. Let me try to visualize the connections again. The figure is A-B-C-D-F-A, with B-F and C-F drawn. So, triangle CDF is formed by points C, D, F. If it's right-angled at D, with angle at D being 45°, then sides CD and DF should be equal. But from coordinates, CD is 6√3 and DF is 6, which are not equal.Wait, maybe the right angle is at F, making CF and DF the legs, and CD the hypotenuse. Then, with angle at D being 45°, the legs should be equal, so CF = DF. But CF is 12, DF is 6, which contradicts.Alternatively, if the right angle is at C, then CD and CF are legs, and DF is the hypotenuse. With angle at D being 45°, the legs should be equal, so CD = CF. But CD is 6√3 ≈10.392, CF is 12, which again contradicts.This is quite confusing. Maybe I need to approach this algebraically without relying on the coordinates.Let me denote:- In triangle ABF: AF = 48, angle at B = 60°, right-angled. So, BF = 24, AB = 24√3.- In triangle BCF: BF = 24, angle at B = 60°, right-angled. So, CF = 12, BC = 12√3.- In triangle CDF: right-angled, angle at D = 45°. So, it's a 45-45-90 triangle. Therefore, legs are equal, hypotenuse is leg√2.Assuming triangle CDF is right-angled at F, then CF = DF, and CD = CF√2. But from triangle BCF, CF = 12, so DF should be 12, and CD should be 12√2.But in the coordinate system, DF is only 6. So, perhaps the figure is scaled down by a factor. Let me see:If CF is 12 in reality, but in the coordinates, it's represented as 12 units, but DF is 6 units. That suggests a scaling factor of 0.5. But then, AF would be 24 in reality, which contradicts AF being 48.Alternatively, maybe the Asymptote code is using a different scale for different parts. This is getting too convoluted.Perhaps I should stick to the given information and ignore the coordinates. The problem states that AF = 48, and the triangles are right-angled with the given angles. So, based on that:1. Triangle ABF: 30-60-90, hypotenuse AF = 48, so BF = 24, AB = 24√3.2. Triangle BCF: 30-60-90, hypotenuse BF = 24, so CF = 12, BC = 12√3.3. Triangle CDF: 45-45-90, so CF = DF, and CD = CF√2. Therefore, CF = 12, DF = 12, CD = 12√2.But in the coordinates, CD is 6√3, which is approximately 10.392, and 12√2 is approximately 16.97. So, unless there's a scaling factor, this doesn't align. But since the problem doesn't mention scaling, I think the coordinates are just for illustration and not to scale.Therefore, based purely on the given information and the properties of the triangles, CF should be 12 units.I think I've spent enough time on this. Despite the coordinate discrepancies, the logical steps based on the triangle properties lead me to conclude that CF is 12.

answer：To rationalize the denominator of the expression (frac{2+sqrt{5}}{2-sqrt{5}}), we can follow these steps:1. **Multiply by the Conjugate**: Multiply both the numerator and the denominator by the conjugate of the denominator, which is (2 + sqrt{5}). This helps to eliminate the square root in the denominator. [ frac{2+sqrt{5}}{2-sqrt{5}} times frac{2+sqrt{5}}{2+sqrt{5}} = frac{(2+sqrt{5})(2+sqrt{5})}{(2-sqrt{5})(2+sqrt{5})} ]2. **Simplify the Denominator**: Use the difference of squares formula to simplify the denominator. [ (2 - sqrt{5})(2 + sqrt{5}) = 2^2 - (sqrt{5})^2 = 4 - 5 = -1 ]3. **Expand the Numerator**: Expand the numerator using the distributive property. [ (2 + sqrt{5})(2 + sqrt{5}) = 2 times 2 + 2 times sqrt{5} + sqrt{5} times 2 + sqrt{5} times sqrt{5} = 4 + 2sqrt{5} + 2sqrt{5} + 5 = 9 + 4sqrt{5} ]4. **Combine the Results**: Now, the expression becomes: [ frac{9 + 4sqrt{5}}{-1} = -9 - 4sqrt{5} ]5. **Identify Coefficients**: The expression (-9 - 4sqrt{5}) is in the form (A + Bsqrt{C}), where (A = -9), (B = -4), and (C = 5).6. **Calculate the Product (ABC)**: [ ABC = (-9) times (-4) times 5 = 180 ]Therefore, the final answer is (boxed{180}).

answer：First, I need to determine the number of grains of rice on the 15th square. Since each square ( k ) contains ( 3^k ) grains, the 15th square has ( 3^{15} ) grains.Next, I need to calculate the total number of grains on the first 10 squares. This is a geometric series where the first term ( a = 3^1 = 3 ) and the common ratio ( r = 3 ). The sum ( S ) of the first 10 terms of a geometric series can be calculated using the formula:[S = a times frac{r^n - 1}{r - 1}]Plugging in the values:[S = 3 times frac{3^{10} - 1}{3 - 1} = 3 times frac{59049 - 1}{2} = 3 times 29524 = 88572]Finally, to find how many more grains are on the 15th square compared to the first 10 squares combined, I subtract the total grains on the first 10 squares from the grains on the 15th square:[3^{15} - S = 14348907 - 88572 = 14260335]