💡First, I need to determine the original price of the coat before the discount. The buyer saved 6.25 and spent 43.75 on the coat. Therefore, the original price can be calculated by adding the amount saved to the amount spent.Next, I'll calculate the percentage saved by dividing the amount saved by the original price and then multiplying by 100 to convert it to a percentage.Finally, I'll compare the calculated percentage to the provided answer choices and select the closest one.

💡Okay, so I've got this function f(x) = (e^x - e^{-x}) / (e^x + e^{-x}), and I need to do two things: first, find the value of f(ln 2), and second, prove that f(x) is an odd function. Hmm, let's start with the first part because it seems more straightforward.Alright, for part (1), I need to compute f(ln 2). That means I should substitute x with ln 2 in the function. So, let's write that out:f(ln 2) = (e^{ln 2} - e^{-ln 2}) / (e^{ln 2} + e^{-ln 2})Hmm, okay. I remember that e^{ln a} is just a, right? Because the natural logarithm and the exponential function are inverses. So, e^{ln 2} should be 2. Similarly, e^{-ln 2} is the same as 1 / e^{ln 2}, which is 1/2. Let me write that down:e^{ln 2} = 2 e^{-ln 2} = 1/2So substituting these back into the function:f(ln 2) = (2 - 1/2) / (2 + 1/2)Now, let's compute the numerator and the denominator separately. The numerator is 2 - 1/2. Well, 2 is the same as 4/2, so 4/2 - 1/2 is 3/2. The denominator is 2 + 1/2, which is 5/2. So now we have:f(ln 2) = (3/2) / (5/2)Dividing fractions is the same as multiplying by the reciprocal, so:(3/2) * (2/5) = (3*2)/(2*5) = 6/10 = 3/5Okay, so f(ln 2) is 3/5. That seems straightforward. I think that's correct. Let me just double-check my steps:1. Substitute x with ln 2.2. Simplify e^{ln 2} to 2 and e^{-ln 2} to 1/2.3. Subtract and add these in numerator and denominator.4. Simplify fractions.Everything seems to check out. So, part (1) is done.Now, moving on to part (2): proving that f(x) is an odd function. I remember that a function is odd if f(-x) = -f(x) for all x in its domain. So, I need to compute f(-x) and see if it equals -f(x).Let's write out f(-x):f(-x) = (e^{-x} - e^{x}) / (e^{-x} + e^{x})Hmm, let's see. Let me factor out a negative sign from the numerator:= (- (e^{x} - e^{-x})) / (e^{-x} + e^{x})So, that's equal to:= - (e^{x} - e^{-x}) / (e^{-x} + e^{x})But wait, the denominator is e^{-x} + e^{x}, which is the same as e^{x} + e^{-x}, just written in reverse order. Since addition is commutative, e^{-x} + e^{x} is equal to e^{x} + e^{-x}. So, the denominator remains the same.Therefore, f(-x) = - (e^{x} - e^{-x}) / (e^{x} + e^{-x}) = -f(x)Which is exactly the definition of an odd function. So, f(-x) = -f(x), hence f(x) is odd.Wait, let me make sure I didn't make any mistakes here. I substituted -x into the function, rearranged the numerator by factoring out a negative, and recognized that the denominator is the same as the original function's denominator. So, yes, f(-x) is indeed equal to -f(x). Therefore, f(x) is an odd function.Is there another way to think about this? Maybe by recognizing that f(x) is similar to the hyperbolic tangent function. Let me recall that tanh(x) = (e^x - e^{-x}) / (e^x + e^{-x}), which is exactly the function f(x) given here. Since tanh(x) is known to be an odd function, that's another way to confirm it. But since the problem asks to prove it, I should stick to the algebraic approach rather than just citing the property of tanh(x).So, to recap:1. Start with f(-x).2. Substitute -x into the function.3. Factor out a negative from the numerator.4. Recognize that the denominator is the same as in f(x).5. Thus, f(-x) = -f(x), proving it's odd.Yes, that seems solid. I don't think I missed anything here.Just to be thorough, let me also consider the domain of f(x). The function f(x) is defined for all real numbers x because e^x and e^{-x} are always positive and never zero, so the denominator e^x + e^{-x} is never zero. Therefore, the domain is all real numbers, which is symmetric about the origin, a necessary condition for a function to be odd. So, that's another checkmark.In summary, both parts seem to be correctly solved. I think I'm confident with these answers.**Final Answer**(1) boxed{dfrac{3}{5}}(2) boxed{text{The function } f(x) text{ is an odd function}}

💡Okay, so I have this problem here about salami circles on a plate. Let me try to figure it out step by step. First, the problem says that eight small salami circles precisely fit across the diameter of a 16-inch plate. Hmm, okay. So if eight circles fit across the diameter, that means each circle has a diameter of 16 inches divided by 8, right? So, 16 divided by 8 is 2. So each salami circle has a diameter of 2 inches. Wait, so if the diameter is 2 inches, then the radius of each salami circle must be half of that, which is 1 inch. Got it. So each salami circle has a radius of 1 inch.Now, the problem also says that a total of 32 circles of salami are placed on this plate without overlapping. I need to find what fraction of the plate is covered by salami. Alright, so to find the fraction, I think I need to calculate the total area covered by the salami circles and then divide that by the total area of the plate. That should give me the fraction of the plate covered by salami.Let me start by calculating the area of one salami circle. The formula for the area of a circle is π times radius squared. So, for one salami circle, that would be π times (1 inch) squared, which is π square inches. Since there are 32 salami circles, the total area covered by all the salami is 32 times π square inches. So, 32π square inches.Next, I need to find the area of the entire plate. The plate has a diameter of 16 inches, so its radius is half of that, which is 8 inches. Using the area formula again, the area of the plate is π times (8 inches) squared, which is π times 64, so 64π square inches.Now, to find the fraction of the plate covered by salami, I divide the total area of the salami circles by the area of the plate. That would be 32π divided by 64π. Wait, the π cancels out, so it's just 32 divided by 64, which simplifies to 1/2. So, the fraction of the plate covered by salami is 1/2. Let me double-check my steps to make sure I didn't make a mistake. 1. Calculated the diameter of each salami circle by dividing the plate's diameter by 8, which gave me 2 inches. 2. Found the radius by halving the diameter, resulting in 1 inch. 3. Calculated the area of one salami circle as π square inches. 4. Multiplied that by 32 to get the total salami area, which is 32π. 5. Calculated the plate's area as 64π square inches. 6. Divided the salami area by the plate area, simplifying to 1/2. Everything seems to add up. I don't think I made any calculation errors. So, yeah, the fraction should be 1/2.

💡Okay, so I have this problem about four flower beds that overlap, and I need to find the total number of unique plants among them. Let me try to break this down step by step.First, let's list out what we know:- Bed A has 600 plants.- Bed B has 550 plants.- Bed C has 400 plants.- Bed D has 300 plants.Now, the overlaps:- Beds A and B share 75 plants.- Beds A and C share 125 plants.- Beds B and D share 50 plants.- Beds A, B, and C share 25 plants together.Hmm, okay. So, we have some overlaps between pairs and even a triple overlap between A, B, and C. I think I need to use the inclusion-exclusion principle here to account for all these overlaps and find the total number of unique plants.Let me recall the inclusion-exclusion principle for four sets. The formula is:[ |A cup B cup C cup D| = |A| + |B| + |C| + |D| - |A cap B| - |A cap C| - |A cap D| - |B cap C| - |B cap D| - |C cap D| + |A cap B cap C| + |A cap B cap D| + |A cap C cap D| + |B cap C cap D| - |A cap B cap C cap D| ]Wow, that's a mouthful. But let's see what we have and what we don't.We know:- |A| = 600- |B| = 550- |C| = 400- |D| = 300- |A ∩ B| = 75- |A ∩ C| = 125- |B ∩ D| = 50- |A ∩ B ∩ C| = 25But we don't have information about |A ∩ D|, |B ∩ C|, |A ∩ C ∩ D|, |B ∩ C ∩ D|, or |A ∩ B ∩ C ∩ D|. Since these aren't provided, I think we have to assume they are zero. That might simplify things.So, plugging in the known values:[ |A cup B cup C cup D| = 600 + 550 + 400 + 300 - 75 - 125 - 0 - 0 - 50 - 0 + 25 + 0 + 0 + 0 - 0 ]Let me compute this step by step.First, add up all the individual beds:600 (A) + 550 (B) + 400 (C) + 300 (D) = 600 + 550 = 1150; 1150 + 400 = 1550; 1550 + 300 = 1850.Now, subtract the pairwise overlaps:1850 - 75 (A ∩ B) = 1775; 1775 - 125 (A ∩ C) = 1650; 1650 - 0 (A ∩ D) = 1650; 1650 - 0 (B ∩ C) = 1650; 1650 - 50 (B ∩ D) = 1600; 1600 - 0 (C ∩ D) = 1600.Now, add back the triple overlaps:1600 + 25 (A ∩ B ∩ C) = 1625; 1625 + 0 (A ∩ B ∩ D) = 1625; 1625 + 0 (A ∩ C ∩ D) = 1625; 1625 + 0 (B ∩ C ∩ D) = 1625.Finally, subtract the quadruple overlap, which is zero.So, the total number of unique plants is 1625.Wait, let me double-check my calculations to make sure I didn't make a mistake.Adding the individual beds: 600 + 550 + 400 + 300 = 1850. That seems right.Subtracting the pairwise overlaps: 75 + 125 + 50 = 250. So, 1850 - 250 = 1600.Adding back the triple overlap: 1600 + 25 = 1625.Yes, that matches. I think I did it correctly.So, the total number of unique plants among these beds is 1625.

💡First, I need to identify the smallest prime numbers that are greater than or equal to 10. The primes in this range are 11, 13, and 17.Next, I calculate the product of these three primes to find the smallest number that has these three different prime factors. Multiplying them together:11 × 13 = 143Then, 143 × 17 = 2431Therefore, the smallest number with three different prime factors, none of which are less than 10, is 2431.

💡Okay, so I have this problem about converting a polar equation to Cartesian coordinates and then finding the sum of distances from a point on the y-axis to two intersection points. Let me try to work through it step by step.First, part (1) asks for the Cartesian equation of curve C, which is given in polar coordinates as ρ = 2(cosθ + sinθ). I remember that in polar coordinates, ρ is the radius, θ is the angle, and to convert to Cartesian, we can use the relationships x = ρcosθ and y = ρsinθ. Also, ρ² = x² + y².So, starting with the given equation: ρ = 2(cosθ + sinθ). Maybe I can multiply both sides by ρ to make it easier to substitute x and y. Let me try that:ρ * ρ = 2ρcosθ + 2ρsinθWhich simplifies to:ρ² = 2ρcosθ + 2ρsinθNow, substituting ρ² with x² + y², and ρcosθ with x, and ρsinθ with y:x² + y² = 2x + 2yHmm, that looks like a circle equation. Maybe I can rearrange it to standard form. Let me move the 2x and 2y to the left side:x² - 2x + y² - 2y = 0To complete the square for both x and y terms. For the x terms: take x² - 2x, complete the square by adding (2/2)² = 1. Similarly for y terms: y² - 2y, add (2/2)² = 1.So, adding 1 and 1 to both sides:x² - 2x + 1 + y² - 2y + 1 = 0 + 1 + 1Which simplifies to:(x - 1)² + (y - 1)² = 2Okay, so that's the Cartesian equation. It's a circle centered at (1,1) with radius √2. That seems right.Now, moving on to part (2). We have a line l given parametrically:x = (1/2)ty = 1 + (√3/2)tWe need to find where this line intersects curve C, which we now know is the circle (x - 1)² + (y - 1)² = 2. The line intersects the circle at points A and B, and it also intersects the y-axis at point E. Then, we have to find |EA| + |EB|.First, let's find the points of intersection A and B. To do that, substitute the parametric equations of the line into the circle equation.Substituting x = (1/2)t and y = 1 + (√3/2)t into (x - 1)² + (y - 1)² = 2:[( (1/2)t - 1 )² + ( (1 + (√3/2)t - 1 )² ] = 2Simplify each term:First term: ( (1/2)t - 1 )² = ( (t/2) - 1 )² = (t/2 - 1)²Second term: (1 + (√3/2)t - 1 )² = ( (√3/2)t )²So, expanding both:First term: (t/2 - 1)² = (t²/4 - t + 1)Second term: ( (√3/2)t )² = (3/4)t²Adding them together:(t²/4 - t + 1) + (3t²/4) = 2Combine like terms:t²/4 + 3t²/4 = (4t²)/4 = t²So, t² - t + 1 = 2Subtract 2 from both sides:t² - t - 1 = 0So, we have a quadratic equation: t² - t - 1 = 0Let me solve for t using the quadratic formula. The quadratic is at² + bt + c = 0, so a = 1, b = -1, c = -1.t = [1 ± √(1 + 4)] / 2 = [1 ± √5]/2So, the two solutions are t = (1 + √5)/2 and t = (1 - √5)/2.These are the parameter values at points A and B. Let me denote t₁ = (1 + √5)/2 and t₂ = (1 - √5)/2.Now, we need to find points A and B. Since the parametric equations are given, we can plug t₁ and t₂ into them.But before that, we need to find point E, which is where the line intersects the y-axis. The y-axis is where x = 0. So, set x = 0 in the parametric equation:x = (1/2)t = 0 => t = 0So, when t = 0, y = 1 + (√3/2)*0 = 1. Therefore, point E is (0, 1).Now, we need to find |EA| and |EB|, then sum them.But since E is on the line l, and A and B are also on l, the distances |EA| and |EB| can be found using the parameter t. Because in parametric equations, the parameter t often represents a scaled distance along the line.But let's think about it. The parametric equations are:x = (1/2)ty = 1 + (√3/2)tSo, the direction vector of the line is (1/2, √3/2). The length of this direction vector is √[(1/2)² + (√3/2)²] = √[1/4 + 3/4] = √[1] = 1. So, the parameter t actually represents the actual distance from the point when t=0, which is E(0,1). So, t is the distance from E.Wait, is that correct? Let me check.If the direction vector is (1/2, √3/2), which has length 1, then yes, t is the actual distance from the point (0,1). So, when t increases by 1, you move 1 unit along the line. Therefore, the parameter t corresponds to the actual distance from E.Therefore, the points A and B correspond to t₁ and t₂, so the distances from E to A and E to B are |t₁| and |t₂|, respectively.But we need to be careful because t can be positive or negative. Let me compute t₁ and t₂:t₁ = (1 + √5)/2 ≈ (1 + 2.236)/2 ≈ 1.618t₂ = (1 - √5)/2 ≈ (1 - 2.236)/2 ≈ -0.618So, t₁ is positive, and t₂ is negative. Since t represents distance from E, the negative t would mean the point is in the opposite direction from E. However, in terms of distance, |EA| is |t₁| and |EB| is |t₂|.But wait, actually, in parametric terms, t can be positive or negative, but the distance from E is the absolute value of t. So, |EA| = |t₁| and |EB| = |t₂|.But let me think again. Since t is a parameter, not necessarily the distance. Wait, earlier I thought that the direction vector has length 1, so t is the distance. But let me verify.The parametric equations are:x = (1/2)ty = 1 + (√3/2)tSo, the direction vector is (1/2, √3/2). The length of this vector is √[(1/2)^2 + (√3/2)^2] = √[1/4 + 3/4] = √1 = 1. So, yes, the direction vector is a unit vector. Therefore, t is the actual distance from the point when t=0, which is E(0,1). So, t is the distance along the line from E.Therefore, when t is positive, we move in one direction, and when t is negative, we move in the opposite direction. So, the points A and B are located at distances |t₁| and |t₂| from E.But in our case, t₁ is positive, so |EA| = t₁, and t₂ is negative, so |EB| = |t₂|.But wait, actually, since t is the parameter, and the points A and B are on the line, regardless of the direction. So, the distance from E to A is |t₁|, and from E to B is |t₂|.But let's compute |EA| + |EB|.Given t₁ = (1 + √5)/2 and t₂ = (1 - √5)/2.Compute |t₁| + |t₂|.t₁ is positive, so |t₁| = t₁ = (1 + √5)/2.t₂ is negative, so |t₂| = |(1 - √5)/2| = (√5 - 1)/2.Therefore, |EA| + |EB| = (1 + √5)/2 + (√5 - 1)/2 = [1 + √5 + √5 - 1]/2 = (2√5)/2 = √5.So, the sum of the distances is √5.Wait, but let me think again. Is this correct? Because sometimes, when dealing with parametric lines and distances, you have to consider the actual Euclidean distance, not just the parameter t.But in this case, since the direction vector is a unit vector, t does correspond to the actual distance from E. So, the distances |EA| and |EB| are indeed |t₁| and |t₂|.Alternatively, we can compute the Euclidean distance between E and A, and E and B, using their coordinates.Let me try that to verify.First, find coordinates of A and B.For point A, t = t₁ = (1 + √5)/2.x = (1/2)t₁ = (1/2)*(1 + √5)/2 = (1 + √5)/4y = 1 + (√3/2)t₁ = 1 + (√3/2)*(1 + √5)/2 = 1 + (√3(1 + √5))/4Similarly, for point B, t = t₂ = (1 - √5)/2.x = (1/2)t₂ = (1/2)*(1 - √5)/2 = (1 - √5)/4y = 1 + (√3/2)t₂ = 1 + (√3/2)*(1 - √5)/2 = 1 + (√3(1 - √5))/4Now, point E is (0,1). Let's compute |EA| and |EB|.Compute |EA|:Coordinates of A: ((1 + √5)/4, 1 + (√3(1 + √5))/4 )Coordinates of E: (0,1)Distance EA:√[ ( (1 + √5)/4 - 0 )² + ( 1 + (√3(1 + √5))/4 - 1 )² ]Simplify:√[ ( (1 + √5)/4 )² + ( (√3(1 + √5))/4 )² ]Factor out 1/4:√[ (1/16)(1 + √5)² + (3/16)(1 + √5)² ]Factor out (1 + √5)² /16:√[ ( (1 + √5)² /16 )(1 + 3) ] = √[ ( (1 + 2√5 + 5) /16 ) *4 ] = √[ (6 + 2√5)/16 *4 ] = √[ (6 + 2√5)/4 ] = √[ (3 + √5)/2 ]Similarly, compute |EB|:Coordinates of B: ((1 - √5)/4, 1 + (√3(1 - √5))/4 )Distance EB:√[ ( (1 - √5)/4 - 0 )² + ( 1 + (√3(1 - √5))/4 - 1 )² ]Simplify:√[ ( (1 - √5)/4 )² + ( (√3(1 - √5))/4 )² ]Factor out 1/4:√[ (1/16)(1 - √5)² + (3/16)(1 - √5)² ]Factor out (1 - √5)² /16:√[ ( (1 - √5)² /16 )(1 + 3) ] = √[ ( (1 - 2√5 + 5) /16 ) *4 ] = √[ (6 - 2√5)/16 *4 ] = √[ (6 - 2√5)/4 ] = √[ (3 - √5)/2 ]Now, compute |EA| + |EB|:√[ (3 + √5)/2 ] + √[ (3 - √5)/2 ]Let me compute this sum.Let me denote a = √[ (3 + √5)/2 ] and b = √[ (3 - √5)/2 ]Compute a + b.Let me square (a + b):(a + b)² = a² + 2ab + b²Compute a² = (3 + √5)/2Compute b² = (3 - √5)/2Compute ab = √[ (3 + √5)/2 ] * √[ (3 - √5)/2 ] = √[ (9 - 5)/4 ] = √[4/4] = √1 = 1Therefore, (a + b)² = (3 + √5)/2 + (3 - √5)/2 + 2*1 = [ (3 + √5 + 3 - √5)/2 ] + 2 = (6/2) + 2 = 3 + 2 = 5Therefore, (a + b)² = 5 => a + b = √5So, |EA| + |EB| = √5, which matches our earlier result.Therefore, the sum of the distances is √5.I think that's consistent. So, both methods give the same result, which is reassuring.So, summarizing:1. The Cartesian equation of curve C is (x - 1)² + (y - 1)² = 2.2. The sum |EA| + |EB| is √5.**Final Answer**(1) The Cartesian equation is boxed{(x-1)^2 + (y-1)^2 = 2}.(2) The sum of the distances is boxed{sqrt{5}}.