answer：Okay, so I have this problem about a rectangular floor that measures (a) by (b) feet. Both (a) and (b) are positive integers, and it's given that (b > a). An artist paints a smaller rectangle on the floor, leaving a 2-foot border all around. The unpainted part is exactly one third of the total area. I need to figure out how many possible ordered pairs ((a, b)) there are.First, let me visualize this. The floor is a rectangle, and the painted area is also a rectangle inside it, but smaller. The border around the painted area is uniformly 2 feet wide on all sides. So, the painted rectangle must be 2 feet away from each edge of the floor.That means the dimensions of the painted rectangle are reduced by 4 feet in both length and width compared to the floor. Because if you take away 2 feet from each side, that's a total of 4 feet from the original dimension. So, the painted rectangle has dimensions ((a - 4)) by ((b - 4)).The total area of the floor is (ab). The area of the painted rectangle is ((a - 4)(b - 4)). The unpainted area is the difference between these two, which is (ab - (a - 4)(b - 4)). According to the problem, this unpainted area is one third of the total area. So, I can set up the equation:[ab - (a - 4)(b - 4) = frac{1}{3}ab]Let me expand the left side to simplify this equation. First, expand ((a - 4)(b - 4)):[(a - 4)(b - 4) = ab - 4a - 4b + 16]So, substituting back into the equation:[ab - (ab - 4a - 4b + 16) = frac{1}{3}ab]Simplify the left side:[ab - ab + 4a + 4b - 16 = frac{1}{3}ab]Which simplifies to:[4a + 4b - 16 = frac{1}{3}ab]To make this equation easier to handle, I can multiply both sides by 3 to eliminate the fraction:[3(4a + 4b - 16) = ab]Calculating the left side:[12a + 12b - 48 = ab]Let me rearrange this equation to bring all terms to one side:[ab - 12a - 12b + 48 = 0]Hmm, this looks like a quadratic in terms of (a) and (b). Maybe I can factor this. Let me try adding 144 to both sides to complete the rectangle or something.Wait, actually, I remember a technique where you can add a constant to both sides to factor the equation. Let me see:[ab - 12a - 12b + 48 = 0]If I add 144 to both sides:[ab - 12a - 12b + 48 + 144 = 144]Simplify the left side:[ab - 12a - 12b + 192 = 144]Wait, that doesn't seem helpful. Maybe I made a mistake in the amount to add. Let me try another approach.I recall that equations of the form (ab + c = k) can sometimes be factored by adding a constant to both sides. Let me try to factor this equation.Looking at (ab - 12a - 12b + 48 = 0), I can rewrite it as:[(a - 12)(b - 12) = 48]Wait, let me check that. If I expand ((a - 12)(b - 12)), I get:[ab - 12a - 12b + 144]But in our equation, the constant term is 48, not 144. So, that approach isn't directly applicable. Maybe I need to adjust it.Alternatively, let me subtract 96 from both sides:[ab - 12a - 12b + 48 - 96 = -96]Wait, that gives:[ab - 12a - 12b - 48 = -96]Which simplifies to:[ab - 12a - 12b = -48]Hmm, not sure if that helps. Maybe I should try a different method.Let me go back to the equation:[ab - 12a - 12b + 48 = 0]I can rewrite this as:[ab - 12a - 12b = -48]Now, add 144 to both sides:[ab - 12a - 12b + 144 = 96]Now, factor the left side:[(a - 12)(b - 12) = 96]Ah, that works! So, now I have:[(a - 12)(b - 12) = 96]Great, now this is a product of two integers equal to 96. Since (a) and (b) are positive integers and (b > a), then (b - 12 > a - 12). Also, both (a - 12) and (b - 12) must be positive integers because (a) and (b) are greater than 12 (since the painted area can't have negative dimensions).So, I need to find all pairs of positive integers ((x, y)) such that (x times y = 96) and (y > x). Then, (a = x + 12) and (b = y + 12).Let me list all the factor pairs of 96:1. (1 times 96)2. (2 times 48)3. (3 times 32)4. (4 times 24)5. (6 times 16)6. (8 times 12)Wait, but since (y > x), I don't need to consider the reverse pairs. So, these are all the possible factor pairs where the second number is larger than the first.Now, let's translate these into possible ((a, b)) pairs:1. (x = 1), (y = 96) → (a = 13), (b = 108)2. (x = 2), (y = 48) → (a = 14), (b = 60)3. (x = 3), (y = 32) → (a = 15), (b = 44)4. (x = 4), (y = 24) → (a = 16), (b = 36)5. (x = 6), (y = 16) → (a = 18), (b = 28)6. (x = 8), (y = 12) → (a = 20), (b = 24)Wait, hold on. I think I made a mistake here. Because earlier, I had the equation ((a - 12)(b - 12) = 96), but let me check if this is correct.Wait, in the earlier steps, I had:[ab - 12a - 12b + 48 = 0]Then, I added 144 to both sides:[ab - 12a - 12b + 144 = 96]Which factors to:[(a - 12)(b - 12) = 96]Yes, that's correct. So, the factor pairs are correct.But wait, in the problem statement, it's given that (b > a). So, in each of these pairs, (b) must be greater than (a). Let me check each pair:1. (a = 13), (b = 108) → (108 > 13) ✔️2. (a = 14), (b = 60) → (60 > 14) ✔️3. (a = 15), (b = 44) → (44 > 15) ✔️4. (a = 16), (b = 36) → (36 > 16) ✔️5. (a = 18), (b = 28) → (28 > 18) ✔️6. (a = 20), (b = 24) → (24 > 20) ✔️All these satisfy (b > a). So, there are 6 possible ordered pairs.Wait, but the answer choices only go up to 6, which is option E. But the answer choices are (A) 2, (B) 3, (C) 4, (D) 5, (E) 6.But wait, let me double-check my earlier steps because the answer seems too high. Maybe I made a mistake in the equation.Going back, the unpainted area is one third of the total area, so the painted area is two thirds. Let me verify the equation:Total area: (ab)Painted area: ((a - 4)(b - 4))Unpainted area: (ab - (a - 4)(b - 4) = frac{1}{3}ab)So, painted area is (frac{2}{3}ab).So, ((a - 4)(b - 4) = frac{2}{3}ab)Wait, earlier, I set up the equation as:[ab - (a - 4)(b - 4) = frac{1}{3}ab]Which is correct because unpainted is one third. So, painted is two thirds.But when I expanded, I had:[ab - (ab - 4a - 4b + 16) = frac{1}{3}ab]Simplifies to:[4a + 4b - 16 = frac{1}{3}ab]Multiply both sides by 3:[12a + 12b - 48 = ab]Rearranged:[ab - 12a - 12b + 48 = 0]Which I then manipulated to:[(a - 12)(b - 12) = 96]Wait, but let me check that step again.Starting from:[ab - 12a - 12b + 48 = 0]If I add 144 to both sides:[ab - 12a - 12b + 48 + 144 = 144]Which is:[ab - 12a - 12b + 192 = 144]Wait, that's not correct because 48 + 144 is 192, but 192 ≠ 144. So, that step was wrong.I think I confused the constant term. Let me try again.Starting from:[ab - 12a - 12b + 48 = 0]Let me write it as:[ab - 12a - 12b = -48]Now, to factor this, I can add 144 to both sides:[ab - 12a - 12b + 144 = 96]Because -48 + 144 = 96.Now, factor the left side:[(a - 12)(b - 12) = 96]Yes, that's correct. So, my earlier step was correct.Therefore, the factor pairs of 96 are as I listed, leading to 6 possible ordered pairs. But the answer choices only go up to 6, which is option E. However, the answer choices are (A) 2, (B) 3, (C) 4, (D) 5, (E) 6.But wait, I think I might have made a mistake in interpreting the problem. Let me check the problem again.The problem says the unpainted part forms a border of width 2 feet around the painted rectangle. So, the painted rectangle is 2 feet away from each edge, meaning the dimensions are reduced by 4 feet in both length and width. So, painted area is ((a - 4)(b - 4)). That seems correct.Total area is (ab), unpainted area is (ab - (a - 4)(b - 4)), which is one third of (ab). So, equation is correct.But when I solved it, I got 6 possible pairs, but the answer choices only go up to 6, which is E. However, the options are A to E, with E being 6. So, maybe 6 is the correct answer.Wait, but let me check if all these pairs satisfy the original condition that the unpainted area is one third.Let me take one example: (a = 13), (b = 108).Total area: (13 times 108 = 1404)Painted area: ((13 - 4) times (108 - 4) = 9 times 104 = 936)Unpainted area: (1404 - 936 = 468)Check if 468 is one third of 1404: (1404 / 3 = 468). Yes, correct.Another example: (a = 14), (b = 60)Total area: (14 times 60 = 840)Painted area: (10 times 56 = 560)Unpainted area: (840 - 560 = 280)Check: (840 / 3 = 280). Correct.Another one: (a = 15), (b = 44)Total area: (15 times 44 = 660)Painted area: (11 times 40 = 440)Unpainted area: (660 - 440 = 220)Check: (660 / 3 = 220). Correct.Similarly, (a = 16), (b = 36):Total area: (16 times 36 = 576)Painted area: (12 times 32 = 384)Unpainted area: (576 - 384 = 192)Check: (576 / 3 = 192). Correct.(a = 18), (b = 28):Total area: (18 times 28 = 504)Painted area: (14 times 24 = 336)Unpainted area: (504 - 336 = 168)Check: (504 / 3 = 168). Correct.Lastly, (a = 20), (b = 24):Total area: (20 times 24 = 480)Painted area: (16 times 20 = 320)Unpainted area: (480 - 320 = 160)Check: (480 / 3 = 160). Correct.So, all 6 pairs satisfy the condition. Therefore, the number of possible ordered pairs ((a, b)) is 6.But wait, the answer choices are (A) 2, (B) 3, (C) 4, (D) 5, (E) 6. So, 6 is an option, E.But the initial problem statement says "positive integers with (b > a)". All my pairs satisfy (b > a), so all 6 are valid.However, I think I might have made a mistake because the answer seems to be 6, but the options include E as 6, so that's fine.Wait, but in the initial problem, the answer was given as B) 3, but that was in the first version. Wait, no, the user provided the problem and then an initial answer, but then asked for a new thought process.Wait, no, the initial problem was presented, and then the user provided a thought process leading to answer B) 3, but then the user asked for a new thought process.Wait, no, looking back, the user provided the problem, then a solution leading to answer B) 3, but then the user says "Please reason step by step, and put your final answer within boxed{}."So, the user is asking for a new thought process, not necessarily following the initial solution.But in my thought process, I arrived at 6 possible pairs, but the answer choices only go up to 6, which is E.Wait, but in the initial solution, the user had:They set up the equation as ((a - 6)(b - 6) = 12), leading to 3 pairs. But in my solution, I have ((a - 12)(b - 12) = 96), leading to 6 pairs.So, there's a discrepancy here. I need to figure out which one is correct.Wait, let me go back to the equation setup.The unpainted area is a border of width 2 feet around the painted rectangle. So, the painted rectangle has dimensions ((a - 4)) and ((b - 4)). Therefore, the area of the painted rectangle is ((a - 4)(b - 4)).Total area is (ab). Unpainted area is (ab - (a - 4)(b - 4)), which is equal to (frac{1}{3}ab).So, the equation is:[ab - (a - 4)(b - 4) = frac{1}{3}ab]Expanding ((a - 4)(b - 4)):[ab - 4a - 4b + 16]So, substituting back:[ab - (ab - 4a - 4b + 16) = frac{1}{3}ab]Simplify:[4a + 4b - 16 = frac{1}{3}ab]Multiply both sides by 3:[12a + 12b - 48 = ab]Rearrange:[ab - 12a - 12b + 48 = 0]Now, to factor this, I can add 144 to both sides:[ab - 12a - 12b + 192 = 144]Wait, that's not correct because 48 + 144 = 192, but 192 ≠ 144. So, that step was wrong.Wait, let me try another approach. Let me write the equation as:[ab - 12a - 12b = -48]Now, add 144 to both sides:[ab - 12a - 12b + 144 = 96]Now, factor the left side:[(a - 12)(b - 12) = 96]Yes, that's correct. So, the product of ((a - 12)) and ((b - 12)) is 96.Now, since (a) and (b) are positive integers and (b > a), then (b - 12 > a - 12). Also, both (a - 12) and (b - 12) must be positive integers because (a) and (b) are greater than 12.So, I need to find all pairs of positive integers ((x, y)) such that (x times y = 96) and (y > x). Then, (a = x + 12) and (b = y + 12).Let me list all the factor pairs of 96:1. (1 times 96)2. (2 times 48)3. (3 times 32)4. (4 times 24)5. (6 times 16)6. (8 times 12)These are all the factor pairs where the second number is larger than the first.Now, translating these into ((a, b)) pairs:1. (x = 1), (y = 96) → (a = 13), (b = 108)2. (x = 2), (y = 48) → (a = 14), (b = 60)3. (x = 3), (y = 32) → (a = 15), (b = 44)4. (x = 4), (y = 24) → (a = 16), (b = 36)5. (x = 6), (y = 16) → (a = 18), (b = 28)6. (x = 8), (y = 12) → (a = 20), (b = 24)All these pairs satisfy (b > a). Therefore, there are 6 possible ordered pairs ((a, b)).But wait, the answer choices only go up to 6, which is option E. So, the correct answer should be E) 6.However, in the initial solution provided by the user, they arrived at 3 pairs by setting ((a - 6)(b - 6) = 12). That seems incorrect because when I set up the equation correctly, I arrived at ((a - 12)(b - 12) = 96), leading to 6 pairs.Therefore, I think the correct answer is 6, which is option E.

answer：Okay, so I have this problem about a parabola with a given vertex and some conditions on its equation. I need to find the smallest possible value of 'a'. Let me try to break this down step by step.First, I remember that the vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. In this case, the vertex is (1/3, -5/9). So, plugging that into the vertex form, the equation becomes:y = a(x - 1/3)^2 - 5/9Now, the problem says the equation is of the form y = ax^2 + bx + c, and we need to find the smallest possible value of 'a' given that a > 0 and the discriminant b^2 - 4ac = 7.So, I need to convert the vertex form into the standard form to find expressions for b and c in terms of a. Let me expand the vertex form:y = a(x - 1/3)^2 - 5/9= a(x^2 - (2/3)x + 1/9) - 5/9= a*x^2 - (2a/3)x + (a/9) - 5/9So, comparing this to y = ax^2 + bx + c, I can see that:b = -2a/3c = (a/9) - 5/9Alright, so now I have expressions for b and c in terms of a. The next condition is that the discriminant b^2 - 4ac = 7. Let me plug in the expressions for b and c into this discriminant.First, let's compute b^2:b^2 = (-2a/3)^2 = (4a^2)/9Next, compute 4ac:4ac = 4a*(a/9 - 5/9) = 4a*( (a - 5)/9 ) = (4a(a - 5))/9So, the discriminant is:b^2 - 4ac = (4a^2)/9 - (4a(a - 5))/9Since both terms have the same denominator, I can combine them:= [4a^2 - 4a(a - 5)] / 9Let me simplify the numerator:4a^2 - 4a(a - 5) = 4a^2 - 4a^2 + 20a = 20aSo, the discriminant becomes:20a / 9 = 7Now, solve for a:20a / 9 = 7Multiply both sides by 9:20a = 63Divide both sides by 20:a = 63/20Hmm, 63 divided by 20 is 3.15. But since the problem asks for the smallest possible value of 'a' and a must be positive, 63/20 seems to be the answer.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Vertex form: y = a(x - 1/3)^2 - 5/9. Correct.2. Expanded to standard form: y = ax^2 - (2a/3)x + (a/9 - 5/9). Correct.3. Identified b = -2a/3 and c = (a/9 - 5/9). Correct.4. Calculated discriminant: b^2 - 4ac = (4a^2)/9 - 4a*(a/9 - 5/9). Correct.5. Simplified numerator: 4a^2 - 4a^2 + 20a = 20a. Correct.6. So, 20a / 9 = 7 => a = 63/20. Correct.Everything seems to check out. So, the smallest possible value of 'a' is 63/20.**Final Answer**The smallest possible value of ( a ) is boxed{dfrac{63}{20}}.

answer：Okay, so I have this problem about a rectangular box. It says the total surface area is 150 square inches, and the sum of the lengths of all its edges is 60 inches. I need to find the sum of the lengths of all its interior diagonals. Hmm, let me think about how to approach this.First, let's recall some basics about rectangular boxes. A rectangular box has three dimensions: length, width, and height. Let's call them ( a ), ( b ), and ( c ) respectively. The problem gives me two pieces of information: the total surface area and the total edge length. I need to use these to find the sum of all interior diagonals. Starting with the total surface area. The surface area ( S ) of a rectangular box is given by the formula:[S = 2(ab + bc + ac)]where ( ab ), ( bc ), and ( ac ) are the areas of the three different faces of the box. The problem states that this total surface area is 150 square inches. So, I can write:[2(ab + bc + ac) = 150]Dividing both sides by 2 to simplify:[ab + bc + ac = 75]Okay, so that's one equation involving ( a ), ( b ), and ( c ).Next, the problem mentions the sum of the lengths of all its edges is 60 inches. A rectangular box has 12 edges: four of each dimension. So, the total edge length ( L ) is:[L = 4(a + b + c)]Given that ( L = 60 ), we can write:[4(a + b + c) = 60]Dividing both sides by 4:[a + b + c = 15]Alright, so now I have two equations:1. ( a + b + c = 15 )2. ( ab + bc + ac = 75 )I need to find the sum of all interior diagonals. An interior diagonal of a rectangular box is the diagonal that goes from one corner of the box to the opposite corner, passing through the interior of the box. Each space diagonal can be calculated using the formula:[d = sqrt{a^2 + b^2 + c^2}]Since there are four such diagonals in a rectangular box, the total sum of all interior diagonals would be:[4d = 4sqrt{a^2 + b^2 + c^2}]So, I need to find ( a^2 + b^2 + c^2 ) to compute this sum.I remember that there's a relationship between ( (a + b + c)^2 ) and ( a^2 + b^2 + c^2 ). Specifically:[(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)]We already know ( a + b + c = 15 ) and ( ab + bc + ac = 75 ). Plugging these into the equation:[15^2 = a^2 + b^2 + c^2 + 2 times 75]Calculating ( 15^2 ):[225 = a^2 + b^2 + c^2 + 150]Subtracting 150 from both sides:[a^2 + b^2 + c^2 = 225 - 150 = 75]So, ( a^2 + b^2 + c^2 = 75 ). Now, plugging this back into the formula for the sum of the interior diagonals:[4sqrt{75}]Simplifying ( sqrt{75} ):[sqrt{75} = sqrt{25 times 3} = 5sqrt{3}]Therefore:[4sqrt{75} = 4 times 5sqrt{3} = 20sqrt{3}]So, the sum of the lengths of all the interior diagonals is ( 20sqrt{3} ) inches.Wait, let me double-check my steps to make sure I didn't make a mistake. 1. I defined ( a ), ( b ), and ( c ) as the sides of the box.2. Calculated the surface area correctly: ( 2(ab + bc + ac) = 150 ) leading to ( ab + bc + ac = 75 ).3. Calculated the total edge length correctly: ( 4(a + b + c) = 60 ) leading to ( a + b + c = 15 ).4. Used the identity ( (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac) ) correctly.5. Plugged in the known values: ( 225 = a^2 + b^2 + c^2 + 150 ), so ( a^2 + b^2 + c^2 = 75 ).6. Calculated the interior diagonal sum as ( 4sqrt{75} = 20sqrt{3} ).Everything seems to check out. I don't see any errors in my calculations or reasoning. So, I'm confident that the answer is ( 20sqrt{3} ).**Final Answer**boxed{C}

answer：First, determine the number of sprigs of mint Anderson would normally use for 12 green tea leaves in the original mud mixture. Since he adds two green tea leaves for every sprig of mint, divide the number of green tea leaves by 2:12 green tea leaves ÷ 2 green tea leaves per sprig of mint = 6 sprigs of mintNext, compare the number of sprigs of mint needed in the new mud to the original mud. In the original mud, Anderson adds 3 sprigs of mint per batch. In the new mud, he needs to add 6 sprigs of mint to achieve the same efficacy. This indicates that the efficacy of the ingredients in the new mud is half as effective as in the original mud.Therefore, the ratio of the efficacy of the ingredients in the new mud to the original mud is 1:2.

answer：First, I need to determine the speed of the stream affecting the motorboat's journey between points A and B.The motorboat's speed in still water is 21 km/h, and the total time for the round trip is 7 hours. The distance between A and B is 72 km.When moving upstream against the stream, the effective speed of the motorboat decreases by the stream's speed, while downstream, the effective speed increases by the stream's speed.Let ( v ) represent the speed of the stream in km/h.The time taken to travel upstream is ( frac{72}{21 - v} ) hours, and the time taken to travel downstream is ( frac{72}{21 + v} ) hours.The sum of these times equals the total journey time:[frac{72}{21 - v} + frac{72}{21 + v} = 7]To solve for ( v ), I'll find a common denominator and simplify the equation:[frac{72(21 + v) + 72(21 - v)}{(21 - v)(21 + v)} = 7]Simplifying the numerator:[frac{72 cdot 21 + 72v + 72 cdot 21 - 72v}{441 - v^2} = 7]The ( 72v ) terms cancel out:[frac{72 cdot 42}{441 - v^2} = 7]Multiplying both sides by ( 441 - v^2 ):[72 cdot 42 = 7(441 - v^2)]Calculating the products:[3024 = 3087 - 7v^2]Rearranging to solve for ( v^2 ):[7v^2 = 3087 - 3024][7v^2 = 63][v^2 = 9]Taking the square root:[v = 3]Therefore, the speed of the stream is 3 km/h.

answer：Okay, so I have this problem here: the function is f(x) = 2 sin(ωx + π/3), where ω is greater than 0. Points A and B are the highest and lowest points on the graph of y = f(x). The distance between A and B is given as 2√5. I need to find f(1).Alright, let me break this down. First, I know that the sine function oscillates between -1 and 1. Since it's multiplied by 2, the function f(x) will oscillate between -2 and 2. So, the maximum value is 2, and the minimum value is -2. That means point A, the highest point, has a y-coordinate of 2, and point B, the lowest point, has a y-coordinate of -2.Now, the distance between A and B is given as 2√5. I need to figure out how this distance relates to the function's properties. Since A and B are the highest and lowest points, they must be separated by half a period, right? Because the sine function goes from maximum to minimum over half a period.Wait, actually, in a sine wave, the distance between a maximum and the next minimum is half a period. But in terms of the graph, the points A and B are not just vertically separated; they also have a horizontal separation. So, the distance between A and B isn't just the vertical distance, which would be 4 (from 2 to -2), but it's the straight-line distance between the two points, considering both their vertical and horizontal differences.So, if I think of points A and B on the graph, they have coordinates (x1, 2) and (x2, -2). The distance between them is given by the distance formula: √[(x2 - x1)^2 + (-2 - 2)^2] = √[(x2 - x1)^2 + (-4)^2] = √[(x2 - x1)^2 + 16]. This distance is equal to 2√5.So, I can set up the equation: √[(x2 - x1)^2 + 16] = 2√5. If I square both sides, I get (x2 - x1)^2 + 16 = 20. Then, subtracting 16 from both sides, (x2 - x1)^2 = 4. Taking the square root, x2 - x1 = ±2. Since x2 is after x1 in the function, we can say x2 - x1 = 2.So, the horizontal distance between A and B is 2 units. But in the sine function, the horizontal distance between a maximum and the next minimum is half the period. So, half the period is 2, which means the full period T is 4.Now, the period of the function f(x) = 2 sin(ωx + π/3) is given by T = 2π / ω. We found that T = 4, so 2π / ω = 4. Solving for ω, we get ω = 2π / 4 = π / 2.Alright, so ω is π/2. Now, the function becomes f(x) = 2 sin((π/2)x + π/3). I need to find f(1). Let's plug x = 1 into the function:f(1) = 2 sin((π/2)(1) + π/3) = 2 sin(π/2 + π/3).Hmm, let's compute π/2 + π/3. To add these, I need a common denominator. π/2 is 3π/6, and π/3 is 2π/6. So, 3π/6 + 2π/6 = 5π/6.So, f(1) = 2 sin(5π/6). I remember that sin(5π/6) is sin(π - π/6) = sin(π/6) = 1/2. But wait, sin(5π/6) is actually positive because it's in the second quadrant. So, sin(5π/6) = 1/2.Therefore, f(1) = 2 * (1/2) = 1.Wait, let me double-check that. So, 5π/6 is 150 degrees, right? Sin(150 degrees) is indeed 1/2. So, yes, that's correct.So, f(1) is 1. That should be the answer.**Final Answer**boxed{1}