💡Alright, so I have this geometry problem here. Let me try to parse it out step by step. The problem says:In triangle ABC, AB = 3, AC = 4, angle BAC = 60 degrees. If P is a point in the plane of triangle ABC such that AP = 2, then the maximum value of vector PB · vector PC is ____.Hmm, okay. So, we have triangle ABC with sides AB = 3, AC = 4, and the angle between them, angle BAC, is 60 degrees. Point P is somewhere in the plane such that the distance from A to P is 2. We need to find the maximum value of the dot product of vectors PB and PC.First, I should probably sketch this triangle to visualize it. Let me imagine triangle ABC with point A at the origin, AB going along the x-axis, and AC making a 60-degree angle with AB. So, point B is at (3, 0), and point C is somewhere in the plane. Let me compute the coordinates of point C.Since AC = 4 and angle BAC is 60 degrees, the coordinates of point C can be found using trigonometry. The x-coordinate will be 4*cos(60°) and the y-coordinate will be 4*sin(60°). Cos(60°) is 0.5, and sin(60°) is (√3)/2. So, point C is at (2, 2√3).So, coordinates:- A: (0, 0)- B: (3, 0)- C: (2, 2√3)Now, point P is somewhere such that AP = 2. So, P lies on a circle centered at A with radius 2. We need to find the maximum value of the dot product of vectors PB and PC.Let me recall that the dot product of two vectors is given by:PB · PC = |PB||PC|cosθwhere θ is the angle between vectors PB and PC.But since we need the maximum value, maybe it's easier to express PB and PC in terms of coordinates and then compute the dot product.Alternatively, maybe using vectors from the origin would be better. Let me assign position vectors to points A, B, C, and P.Let me denote:- Vector A as the origin, so vector A = (0, 0)- Vector B = (3, 0)- Vector C = (2, 2√3)- Vector P = (x, y), with the condition that |P - A| = 2, so sqrt(x² + y²) = 2, which means x² + y² = 4.So, vector PB = B - P = (3 - x, 0 - y) = (3 - x, -y)Vector PC = C - P = (2 - x, 2√3 - y)Then, the dot product PB · PC is:(3 - x)(2 - x) + (-y)(2√3 - y)Let me compute that:First term: (3 - x)(2 - x) = 6 - 3x - 2x + x² = x² - 5x + 6Second term: (-y)(2√3 - y) = -2√3 y + y²So, adding both terms:x² - 5x + 6 - 2√3 y + y²So, the dot product PB · PC = x² + y² - 5x - 2√3 y + 6But we know that x² + y² = 4, since P lies on the circle of radius 2 centered at A.So, substituting x² + y² = 4 into the expression:PB · PC = 4 - 5x - 2√3 y + 6 = 10 - 5x - 2√3 ySo, now, the problem reduces to finding the maximum value of the expression 10 - 5x - 2√3 y, given that x² + y² = 4.So, we need to maximize 10 - 5x - 2√3 y, subject to x² + y² = 4.This is a constrained optimization problem. I can use the method of Lagrange multipliers, but since it's a circle, maybe it's easier to parametrize the point P on the circle.Let me parametrize P as (2 cosθ, 2 sinθ), since it's on the circle of radius 2.So, x = 2 cosθ, y = 2 sinθ.Substituting into the expression:10 - 5*(2 cosθ) - 2√3*(2 sinθ) = 10 - 10 cosθ - 4√3 sinθSo, we need to maximize 10 - 10 cosθ - 4√3 sinθ.So, the expression to maximize is 10 - 10 cosθ - 4√3 sinθ.Let me denote the expression as E = 10 - 10 cosθ - 4√3 sinθ.To find the maximum of E, since 10 is a constant, we can focus on minimizing the expression 10 cosθ + 4√3 sinθ.Wait, because E = 10 - (10 cosθ + 4√3 sinθ). So, to maximize E, we need to minimize 10 cosθ + 4√3 sinθ.Alternatively, since E is 10 minus something, the maximum of E occurs when that something is minimized.So, let me consider the expression 10 cosθ + 4√3 sinθ.This is of the form A cosθ + B sinθ, which can be rewritten as R cos(θ - φ), where R = sqrt(A² + B²) and tanφ = B/A.So, let's compute R:R = sqrt(10² + (4√3)²) = sqrt(100 + 16*3) = sqrt(100 + 48) = sqrt(148) = sqrt(4*37) = 2√37.So, 10 cosθ + 4√3 sinθ = 2√37 cos(θ - φ), where φ = arctan((4√3)/10) = arctan((2√3)/5).Therefore, the minimum value of 10 cosθ + 4√3 sinθ is -2√37, because the cosine function ranges between -1 and 1.So, the minimum value is -2√37, which occurs when cos(θ - φ) = -1, i.e., when θ - φ = π, so θ = φ + π.Therefore, the maximum value of E is 10 - (-2√37) = 10 + 2√37.Hence, the maximum value of PB · PC is 10 + 2√37.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I found coordinates for points A, B, C correctly.2. Expressed vectors PB and PC correctly in terms of coordinates.3. Calculated the dot product correctly, expanded it, substituted x² + y² = 4.4. Expressed the problem as maximizing 10 - 5x - 2√3 y with x² + y² = 4.5. Parametrized P as (2 cosθ, 2 sinθ), substituted into the expression, leading to E = 10 - 10 cosθ - 4√3 sinθ.6. Recognized that to maximize E, we need to minimize 10 cosθ + 4√3 sinθ.7. Converted 10 cosθ + 4√3 sinθ into R cos(θ - φ), found R = 2√37, hence the minimum is -2√37.8. Therefore, maximum E is 10 + 2√37.Seems solid. Alternatively, another approach is to use vectors and express PB · PC in terms of vectors from A.Let me try that approach to verify.Let me denote vectors:Let’s set A as the origin, so position vectors:- Vector AB = (3, 0)- Vector AC = (2, 2√3)- Vector AP = (x, y), with |AP| = 2, so x² + y² = 4.Then, vectors PB = AB - AP = (3 - x, 0 - y)Vectors PC = AC - AP = (2 - x, 2√3 - y)Dot product PB · PC = (3 - x)(2 - x) + (-y)(2√3 - y)Which is the same as before, leading to the same expression.Alternatively, maybe using vector properties.Let me recall that for any point P, PB · PC can be expressed in terms of PA, AB, and AC.Wait, perhaps using the formula:PB · PC = (PA + AB) · (PA + AC)Wait, is that correct?Wait, no. Let me think.Wait, vectors PB = B - P = AB - APSimilarly, PC = C - P = AC - APSo, PB · PC = (AB - AP) · (AC - AP) = AB · AC - AB · AP - AC · AP + AP · APWhich is AB · AC - (AB + AC) · AP + |AP|²We know that |AP| = 2, so |AP|² = 4.Also, AB · AC can be computed as |AB||AC|cos(angle BAC) = 3*4*cos60° = 12*(0.5) = 6.So, AB · AC = 6.Therefore, PB · PC = 6 - (AB + AC) · AP + 4 = 10 - (AB + AC) · APSo, PB · PC = 10 - (AB + AC) · APSo, to maximize PB · PC, we need to minimize (AB + AC) · AP.So, let me compute vector AB + AC.Vector AB is (3, 0), vector AC is (2, 2√3). So, AB + AC = (5, 2√3).So, (AB + AC) · AP = (5, 2√3) · (x, y) = 5x + 2√3 y.So, we have PB · PC = 10 - (5x + 2√3 y)Which is the same as before.So, to maximize PB · PC, we need to minimize 5x + 2√3 y, given that x² + y² = 4.So, again, we can use the same method as before.Express 5x + 2√3 y as a dot product of (5, 2√3) and (x, y). The minimum value of this expression is equal to -|(5, 2√3)| * |(x, y)|, by the Cauchy-Schwarz inequality.Compute |(5, 2√3)| = sqrt(25 + (4*3)) = sqrt(25 + 12) = sqrt(37).|(x, y)| = 2, since AP = 2.Therefore, the minimum value of 5x + 2√3 y is -sqrt(37)*2 = -2√37.Therefore, the maximum value of PB · PC is 10 - (-2√37) = 10 + 2√37.Same result as before. So, that confirms the answer.Alternatively, another way is to use the method of Lagrange multipliers.Let me set up the function to maximize:f(x, y) = 10 - 5x - 2√3 ySubject to the constraint g(x, y) = x² + y² - 4 = 0.The method of Lagrange multipliers says that at the extremum, the gradient of f is proportional to the gradient of g.So, compute gradients:∇f = (-5, -2√3)∇g = (2x, 2y)So, ∇f = λ ∇gTherefore:-5 = 2x λ-2√3 = 2y λSo, from the first equation: λ = -5/(2x)From the second equation: λ = -2√3/(2y) = -√3 / yTherefore, -5/(2x) = -√3 / y => 5/(2x) = √3 / y => 5y = 2√3 x => y = (2√3 /5) xNow, substitute y = (2√3 /5) x into the constraint x² + y² = 4:x² + ( (2√3 /5) x )² = 4Compute:x² + (4*3 /25) x² = 4x² + (12/25) x² = 4(25/25 + 12/25) x² = 4(37/25) x² = 4x² = (4 *25)/37 = 100/37So, x = ±10/√37Then, y = (2√3 /5) x = (2√3 /5)(±10/√37) = ±(20√3)/(5√37) = ±(4√3)/√37So, we have two critical points:P1: (10/√37, 4√3 /√37)P2: (-10/√37, -4√3 /√37)Now, compute f(x, y) = 10 -5x -2√3 y at these points.First, at P1:f = 10 -5*(10/√37) -2√3*(4√3 /√37)Compute each term:5*(10/√37) = 50/√372√3*(4√3 /√37) = 2*4*(√3*√3)/√37 = 8*3 /√37 = 24/√37So, f = 10 - 50/√37 -24/√37 = 10 - (50 +24)/√37 = 10 -74/√37Wait, but 74/√37 is 74√37 /37 = 2√37.So, f = 10 -2√37At P2:f = 10 -5*(-10/√37) -2√3*(-4√3 /√37)Compute each term:-5*(-10/√37) = 50/√37-2√3*(-4√3 /√37) = 8*3 /√37 =24/√37So, f = 10 +50/√37 +24/√37 = 10 +74/√37 = 10 +2√37Therefore, the maximum value is 10 +2√37, and the minimum is 10 -2√37.So, this method also confirms the result.Therefore, the maximum value of PB · PC is 10 +2√37.I think that's thorough enough. I considered multiple approaches: coordinate geometry, vector algebra, and Lagrange multipliers, all leading to the same conclusion. So, I'm confident that the answer is 10 +2√37.**Final Answer**The maximum value of overset{→}{PB}cdot overset{→}{PC} is boxed{10 + 2sqrt{37}}.

💡Okay, so I have this equation here: 9 minus 4 divided by x equals 7 plus 8 divided by x. Hmm, that sounds a bit tricky, but I think I can figure it out. Let me write it down to visualize it better: 9 - (4/x) = 7 + (8/x). Alright, so I need to find the value of x that makes this equation true.First, I should probably get all the terms involving x on one side and the constant numbers on the other side. That way, it might be easier to solve for x. So, let's see. On the left side, I have 9 minus (4/x), and on the right side, I have 7 plus (8/x). Maybe I can subtract 7 from both sides to get rid of that 7 on the right. Let me try that.Subtracting 7 from both sides: 9 - 7 - (4/x) = (8/x). That simplifies to 2 - (4/x) = (8/x). Okay, now I have 2 minus (4/x) equals (8/x). Hmm, I still have x in the denominators, which is making it a bit complicated. Maybe I can get rid of the denominators by multiplying both sides by x. That should help eliminate the fractions.Multiplying both sides by x: x*(2 - 4/x) = x*(8/x). Let's distribute the x on the left side: 2x - 4 = 8. Now, that looks simpler. So, 2x minus 4 equals 8. I can solve for x from here. Let's add 4 to both sides to isolate the term with x.Adding 4 to both sides: 2x - 4 + 4 = 8 + 4, which simplifies to 2x = 12. Now, to find x, I just need to divide both sides by 2.Dividing both sides by 2: 2x/2 = 12/2, so x = 6. Wait, does that make sense? Let me check my steps to make sure I didn't make a mistake.Starting from the original equation: 9 - (4/x) = 7 + (8/x). I subtracted 7 from both sides to get 2 - (4/x) = (8/x). Then I multiplied both sides by x to eliminate the denominators, resulting in 2x - 4 = 8. Adding 4 to both sides gave me 2x = 12, and dividing by 2 gave x = 6.Let me plug x = 6 back into the original equation to verify. On the left side: 9 - (4/6) = 9 - (2/3) = 8 and 1/3. On the right side: 7 + (8/6) = 7 + (4/3) = 8 and 1/3. Yes, both sides equal 8 and 1/3, so x = 6 is the correct solution.I think I've got it right. It's always good to check my work to make sure I didn't make any calculation errors along the way. Everything seems to add up, so I'm confident that x = 6 is the correct answer.

💡Alright, so I have this problem about counting the number of 5-card hands that contain four cards of the same value. Let me try to wrap my head around it step by step.First off, the deck has 52 cards, with 13 different values and 4 suits. Each value has one card per suit, so for any given value, there are 4 cards in the deck. A hand is just any combination of 5 cards, and the order doesn't matter.Okay, so I need to find how many different 5-card hands have four cards of the same value. That means in the hand, there are four cards all sharing the same value, like four Aces or four Kings, and then one more card that's different.Let me break this down. First, I need to choose which value is going to be the one that appears four times. There are 13 possible values (1 through 10, Jack, Queen, King). So, I have 13 choices for this.Once I've chosen the value, I need to select all four suits for that value. Since there are exactly four suits, there's only one way to choose all four cards of that value. So, no additional choices here.Now, the fifth card has to be different from the four I've already chosen. Since I've already used up four cards of one value, there are 52 - 4 = 48 cards left in the deck that are of different values. So, I have 48 choices for the fifth card.Putting this all together, the total number of such hands should be the number of choices for the value times the number of choices for the fifth card. That would be 13 * 48.Let me do the multiplication: 13 * 48. Hmm, 10 * 48 is 480, and 3 * 48 is 144, so 480 + 144 is 624.Wait, does this make sense? Let me double-check. I have 13 values, and for each, I can pair it with any of the remaining 48 cards. Yeah, that seems right. I don't think I'm overcounting because each hand is uniquely determined by the value of the four cards and the fifth card.Is there another way to think about this? Maybe using combinations. The number of ways to choose four cards of the same value is C(4,4) for each value, which is 1, and then the number of ways to choose the fifth card is C(48,1). So, total is 13 * 1 * 48 = 624. Yep, that matches.I think that's solid. So, the answer should be 624 different 5-card hands that contain four cards of the same value.

💡Alright, so I have this problem about an acute triangle ABC. The sides opposite angles A, B, and C are a, b, and c respectively. I know that a is given as 2, and there's this equation: b + c equals (4 divided by the square root of 3) times (sin B plus sin C). I need to find angle A and then the maximum area of triangle ABC.First, I remember that in any triangle, the Law of Sines relates the sides and angles. The Law of Sines says that a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle. So, maybe I can use this to express b and c in terms of sine functions.Given that a = 2, I can write 2/sin A = 2R, which simplifies to R = 1/sin A. So, R is related to angle A. Now, since b = 2R sin B and c = 2R sin C, I can substitute R into these expressions. That gives me b = (2/sin A) sin B and c = (2/sin A) sin C.Now, looking at the given equation: b + c = (4/√3)(sin B + sin C). If I substitute my expressions for b and c into this equation, I get:(2/sin A)(sin B + sin C) = (4/√3)(sin B + sin C)Hmm, interesting. Both sides have (sin B + sin C). Assuming that sin B + sin C is not zero, which makes sense because in a triangle, angles B and C can't be zero, so their sines are positive. So, I can divide both sides by (sin B + sin C), which gives:2/sin A = 4/√3Solving for sin A, I get:sin A = (2 * √3)/4 = √3/2Okay, so sin A is √3/2. I know that sin 60° is √3/2, and since the triangle is acute, angle A must be 60°, which is π/3 radians. So, that answers part (I): angle A is π/3.Now, moving on to part (II), finding the maximum area of triangle ABC. The area of a triangle can be calculated using the formula (1/2)ab sin C, but in this case, I can use the formula involving two sides and the included angle. Since I know angle A is π/3, the area S can be written as:S = (1/2) * b * c * sin ASince sin A is √3/2, this becomes:S = (1/2) * b * c * (√3/2) = (√3/4) * b * cSo, to maximize the area, I need to maximize the product b * c.I also know from the Law of Cosines that:a² = b² + c² - 2bc cos AGiven that a = 2 and angle A is π/3, cos A is 0.5. Plugging these into the equation:2² = b² + c² - 2bc * 0.5Simplifying:4 = b² + c² - bcSo, b² + c² - bc = 4I need to find the maximum value of bc given this constraint. Maybe I can express this in terms of bc. Let's rearrange the equation:b² + c² = bc + 4I also know that for any two positive numbers b and c, the arithmetic mean is greater than or equal to the geometric mean. So, (b² + c²)/2 ≥ (bc). But I'm not sure if that's directly helpful here.Alternatively, I can consider using the method of Lagrange multipliers or substitution to maximize bc under the constraint b² + c² - bc = 4. But since this is a triangle, perhaps there's a trigonometric approach.Wait, another idea: let's express b and c in terms of angles B and C. From the Law of Sines, since a = 2 and angle A is π/3, we have:b = (2 / sin A) * sin B = (2 / (√3/2)) * sin B = (4/√3) sin BSimilarly, c = (4/√3) sin CSince in a triangle, angles A + B + C = π, and angle A is π/3, so B + C = 2π/3.Let’s denote angle B as x, so angle C is 2π/3 - x.So, b = (4/√3) sin xc = (4/√3) sin(2π/3 - x)Now, the product bc is:bc = (4/√3)^2 sin x sin(2π/3 - x) = (16/3) sin x sin(2π/3 - x)Let’s compute sin x sin(2π/3 - x). Using the identity:sin A sin B = [cos(A - B) - cos(A + B)] / 2So, sin x sin(2π/3 - x) = [cos(x - (2π/3 - x)) - cos(x + (2π/3 - x))]/2Simplify the arguments:First term: x - (2π/3 - x) = 2x - 2π/3Second term: x + (2π/3 - x) = 2π/3So, it becomes:[cos(2x - 2π/3) - cos(2π/3)] / 2We know that cos(2π/3) is -1/2. So,[cos(2x - 2π/3) - (-1/2)] / 2 = [cos(2x - 2π/3) + 1/2] / 2 = (1/2)cos(2x - 2π/3) + 1/4Therefore, bc = (16/3) * [ (1/2)cos(2x - 2π/3) + 1/4 ] = (16/3)( (1/2)cos(2x - 2π/3) + 1/4 )Simplify:= (16/3)(1/2 cos(2x - 2π/3) + 1/4) = (8/3)cos(2x - 2π/3) + 4/3To maximize bc, we need to maximize cos(2x - 2π/3). The maximum value of cosine is 1, so the maximum bc is:(8/3)(1) + 4/3 = 12/3 = 4So, the maximum value of bc is 4.Therefore, the maximum area S is:S = (√3 / 4) * bc = (√3 / 4) * 4 = √3So, the maximum area is √3.Wait, let me double-check my steps. When I expressed bc in terms of x, I used the identity correctly, right? Yes, I think so. Then, I found that bc is a function of cos(2x - 2π/3), which has a maximum of 1. So, plugging that in gives bc = 4. Then, the area is (√3/4)*4 = √3. That seems correct.Alternatively, I could have used the AM-GM inequality or other methods, but this trigonometric approach seems solid. So, I think the maximum area is indeed √3.**Final Answer**(I) The measure of angle ( A ) is ( boxed{dfrac{pi}{3}} ).(II) The maximum area of triangle ( ABC ) is ( boxed{sqrt{3}} ).

💡Okay, so I have this problem where an alien says it has gathered 321 base 7 units of energy, and I need to figure out what that is in base 10. Hmm, base conversions can be tricky, but I think I can handle this. Let me start by recalling what base 7 means. In base 7, each digit represents a power of 7, just like in base 10 each digit represents a power of 10. So, the rightmost digit is 7^0, the middle digit is 7^1, and the leftmost digit is 7^2, and so on.Alright, so the number given is 321 in base 7. That means there are three digits: 3, 2, and 1. I need to convert each of these digits into their respective place values in base 10. Let me write this out step by step.First, let's identify the place values. The rightmost digit is the units place, which is 7^0. The middle digit is the sevens place, which is 7^1. The leftmost digit is the forty-nines place, which is 7^2. So, breaking it down:- The digit '3' is in the 7^2 place.- The digit '2' is in the 7^1 place.- The digit '1' is in the 7^0 place.Now, I need to calculate the value of each digit multiplied by its respective power of 7.Starting with the leftmost digit: 3 multiplied by 7 squared. Let me compute 7 squared first. 7 times 7 is 49. So, 3 times 49 equals... let's see, 3 times 40 is 120, and 3 times 9 is 27, so 120 plus 27 is 147. Okay, so the first part is 147.Next, the middle digit: 2 multiplied by 7 to the power of 1. Well, 7 to the first power is just 7. So, 2 times 7 is 14. That seems straightforward.Finally, the rightmost digit: 1 multiplied by 7 to the power of 0. Any number to the power of 0 is 1, so 1 times 1 is 1.Now, I have three values: 147, 14, and 1. To get the total in base 10, I need to add these together. Let's do that step by step.First, add 147 and 14. 147 plus 14 is... let's see, 147 plus 10 is 157, and then plus 4 more is 161. So, that gives me 161.Then, I need to add the last value, which is 1. So, 161 plus 1 equals 162.Wait a second, let me double-check my calculations to make sure I didn't make any mistakes. Starting with 3 times 49: 3 times 40 is 120, and 3 times 9 is 27, so 120 plus 27 is indeed 147. Then, 2 times 7 is 14, and 1 times 1 is 1. Adding them up: 147 plus 14 is 161, and 161 plus 1 is 162. Yeah, that seems correct.Just to be thorough, let me try another way of adding them. Maybe add 147 and 1 first, and then add 14. So, 147 plus 1 is 148, and then 148 plus 14 is 162. Yep, same result. Okay, so that checks out.I think I've got it. The number 321 in base 7 converts to 162 in base 10. It makes sense because each digit is contributing its value multiplied by increasing powers of 7, and when you sum them all up, you get the total in base 10.Just to make sure I understand the concept correctly, let me think about why we use powers of 7. In any base system, the place values are powers of that base. So, in base 7, the first place is 7^0, the second is 7^1, the third is 7^2, and so on. This is similar to how in base 10, the places are 10^0, 10^1, 10^2, etc.So, applying that to 321 base 7, it's like saying 3 times 7^2 plus 2 times 7^1 plus 1 times 7^0. Calculating each term individually and then summing them gives the base 10 equivalent.I wonder if there's a quicker way to do this without breaking it down into separate terms. Maybe by using a formula or a shortcut? Let me think. Well, the general formula for converting a number from base b to base 10 is to take each digit, multiply it by b raised to the power of its position (starting from 0 on the right), and then sum all those products. So, in this case, it's exactly what I did: 3*7^2 + 2*7^1 + 1*7^0.I guess there's no real shortcut other than being careful with the exponents and the multiplication. It's all about understanding the place values in the given base and translating them appropriately into base 10.Another thing to consider is whether the digits in the original number are valid for base 7. In base 7, the digits can only range from 0 to 6. Looking at 321 base 7, the digits are 3, 2, and 1, all of which are less than 7. So, that's good; there's no invalid digit here that would make the number impossible in base 7.If there were a digit equal to or larger than 7, that would be a problem because base 7 digits can't exceed 6. For example, if the number was 381 base 7, the digit '8' would be invalid since 8 is greater than 7. But in this case, all digits are valid, so the conversion is straightforward.I also recall that when converting from a higher base to base 10, it's essential to handle each digit correctly and not mix up the exponents. For instance, if I mistakenly used 7^3 instead of 7^2 for the leftmost digit, I would get an incorrect result. So, paying attention to the position of each digit is crucial.Let me try to visualize the number 321 base 7 as a sum of its components:- The '3' is in the 49s place, so it's worth 3*49 = 147.- The '2' is in the 7s place, so it's worth 2*7 = 14.- The '1' is in the 1s place, so it's worth 1*1 = 1.Adding these together: 147 + 14 + 1 = 162.This breakdown helps me see how each digit contributes to the overall value in base 10. It's like breaking down a complex number into simpler, more manageable parts.I also think about why bases are important in different contexts. For example, computers use base 2 (binary) because it's easy to represent with electrical signals (on/off). Similarly, base 16 (hexadecimal) is used in computing because it's a more compact way to represent binary numbers. Understanding different bases is fundamental in fields like computer science and engineering.In this problem, though, it's just a simple base conversion, but it reinforces the importance of place value and how numbers are represented differently across various numeral systems.To ensure I've fully grasped the concept, let me try converting another base 7 number to base 10 on my own. Suppose I have the number 10 base 7. Converting that to base 10 would be:- The '1' is in the 7^1 place, so 1*7 = 7.- The '0' is in the 7^0 place, so 0*1 = 0.- Adding them together: 7 + 0 = 7.So, 10 base 7 is 7 in base 10. That makes sense because 10 in any base b is equal to b in base 10. Similarly, 100 base 7 would be 49 in base 10, since 1*7^2 + 0*7^1 + 0*7^0 = 49 + 0 + 0 = 49.Going back to the original problem, 321 base 7, I feel confident that my conversion to 162 base 10 is accurate. I've double-checked my calculations, ensured that all digits are valid for base 7, and even tested the concept with a simpler number to confirm my understanding.In summary, converting a number from base 7 to base 10 involves multiplying each digit by 7 raised to the power of its position (starting from 0 on the right) and then summing those products. Applying this method to 321 base 7 gives us 162 in base 10.

💡Alright, so I've got this trigonometry problem here, and I need to figure out the value of cos(2α + π/3) given that sin(α - π/3) = √3/3. Hmm, okay. Let me break this down step by step.First, I remember that there are some trigonometric identities that relate sine and cosine functions, especially when dealing with angles that are shifted or scaled. Since the problem gives me a sine function and asks for a cosine function, maybe I can use some kind of identity to connect them.I recall that the double-angle formulas might be useful here. Specifically, the formula for cos(2θ) is 1 - 2sin²θ or 2cos²θ - 1. Maybe I can apply that somehow. But wait, the angle inside the cosine function isn't just 2α; it's 2α + π/3. That complicates things a bit.Let me write down what I know:sin(α - π/3) = √3/3.I need to find cos(2α + π/3). Hmm, perhaps I can express 2α + π/3 in terms of α - π/3. Let me see:2α + π/3 = 2(α - π/3) + 2π/3 + π/3 = 2(α - π/3) + π.Wait, that doesn't seem right. Let me check that again.If I take 2α + π/3 and try to express it in terms of (α - π/3), let's see:Let me set θ = α - π/3. Then, α = θ + π/3.Substituting back into 2α + π/3:2α + π/3 = 2(θ + π/3) + π/3 = 2θ + 2π/3 + π/3 = 2θ + π.So, 2α + π/3 = 2θ + π, where θ = α - π/3.Therefore, cos(2α + π/3) = cos(2θ + π).Now, I remember that cos(φ + π) = -cosφ, because adding π to an angle reflects it across the x-axis, changing the sign of the cosine value.So, cos(2θ + π) = -cos(2θ).Therefore, cos(2α + π/3) = -cos(2θ) = -cos(2(α - π/3)).Now, I can use the double-angle formula for cosine:cos(2θ) = 1 - 2sin²θ.Since θ = α - π/3, and we know sin(θ) = √3/3, we can substitute that in:cos(2θ) = 1 - 2*(√3/3)².Let's compute that:(√3/3)² = (3)/(9) = 1/3.So, cos(2θ) = 1 - 2*(1/3) = 1 - 2/3 = 1/3.Therefore, cos(2α + π/3) = -cos(2θ) = -1/3.Wait, so that would mean the answer is -1/3, which corresponds to option B.But let me double-check my steps to make sure I didn't make any mistakes.1. I set θ = α - π/3, so α = θ + π/3.2. Then, 2α + π/3 becomes 2θ + 2π/3 + π/3 = 2θ + π.3. So, cos(2α + π/3) = cos(2θ + π) = -cos(2θ).4. Using the double-angle formula: cos(2θ) = 1 - 2sin²θ.5. Substituting sinθ = √3/3: cos(2θ) = 1 - 2*(1/3) = 1 - 2/3 = 1/3.6. Therefore, cos(2α + π/3) = -1/3.Everything seems to check out. I don't see any errors in the reasoning.Alternatively, I could have used another identity or approach, but this seems straightforward and leads me to the answer.Just to explore another method, maybe I can express cos(2α + π/3) directly in terms of sin(α - π/3). Let's see:First, let's note that 2α + π/3 can be written as 2(α - π/3) + 2π/3 + π/3 = 2(α - π/3) + π, which is similar to what I did before.Alternatively, maybe I can use the cosine of a sum formula:cos(A + B) = cosA cosB - sinA sinB.But in this case, A = 2α and B = π/3. So,cos(2α + π/3) = cos(2α)cos(π/3) - sin(2α)sin(π/3).I know that cos(π/3) = 1/2 and sin(π/3) = √3/2.So, cos(2α + π/3) = (1/2)cos(2α) - (√3/2)sin(2α).Now, I need expressions for cos(2α) and sin(2α). Maybe I can find these using the given information.Given that sin(α - π/3) = √3/3, let's denote θ = α - π/3, so sinθ = √3/3.From this, I can find cosθ using the Pythagorean identity:sin²θ + cos²θ = 1.So, cos²θ = 1 - sin²θ = 1 - (1/3) = 2/3.Therefore, cosθ = ±√(2/3) = ±√6/3.Now, depending on the quadrant where θ lies, cosθ could be positive or negative. But since the problem doesn't specify the quadrant, I might have to consider both possibilities or see if it affects the final answer.But let's proceed.We have θ = α - π/3, so α = θ + π/3.Therefore, 2α = 2θ + 2π/3.So, cos(2α) = cos(2θ + 2π/3).Similarly, sin(2α) = sin(2θ + 2π/3).Again, using the sum formulas:cos(2θ + 2π/3) = cos(2θ)cos(2π/3) - sin(2θ)sin(2π/3).Similarly, sin(2θ + 2π/3) = sin(2θ)cos(2π/3) + cos(2θ)sin(2π/3).We know that cos(2π/3) = -1/2 and sin(2π/3) = √3/2.So, let's compute cos(2α) and sin(2α):First, compute cos(2θ):cos(2θ) = 1 - 2sin²θ = 1 - 2*(1/3) = 1 - 2/3 = 1/3.Similarly, sin(2θ) = 2sinθcosθ = 2*(√3/3)*(±√6/3) = ±2*(√18)/9 = ±2*(3√2)/9 = ±(2√2)/3.Wait, hold on. Let's compute sin(2θ):sin(2θ) = 2sinθcosθ.We have sinθ = √3/3 and cosθ = ±√6/3.So, sin(2θ) = 2*(√3/3)*(±√6/3) = ±2*(√18)/9.Simplify √18: √18 = 3√2.So, sin(2θ) = ±2*(3√2)/9 = ±(6√2)/9 = ±(2√2)/3.Okay, so sin(2θ) is either 2√2/3 or -2√2/3, depending on the sign of cosθ.But since we don't know the quadrant of θ, we might have to consider both cases. However, let's see if the final answer is affected by this.Now, let's compute cos(2α):cos(2α) = cos(2θ + 2π/3) = cos(2θ)cos(2π/3) - sin(2θ)sin(2π/3).Substituting the known values:cos(2θ) = 1/3, cos(2π/3) = -1/2, sin(2θ) = ±2√2/3, sin(2π/3) = √3/2.So,cos(2α) = (1/3)*(-1/2) - (±2√2/3)*(√3/2).Simplify:= (-1/6) - (±2√6)/6.Similarly, sin(2α) = sin(2θ + 2π/3) = sin(2θ)cos(2π/3) + cos(2θ)sin(2π/3).Substituting:sin(2α) = (±2√2/3)*(-1/2) + (1/3)*(√3/2).Simplify:= (±(-2√2)/6) + (√3)/6.= (±(-√2)/3) + (√3)/6.Now, going back to cos(2α + π/3):cos(2α + π/3) = (1/2)cos(2α) - (√3/2)sin(2α).Substituting the expressions for cos(2α) and sin(2α):= (1/2)[(-1/6) - (±2√6)/6] - (√3/2)[(±(-√2)/3) + (√3)/6].Let's compute each term step by step.First term: (1/2)[(-1/6) - (±2√6)/6] = (1/2)*(-1/6) - (1/2)*(±2√6)/6 = (-1/12) - (±√6)/6.Second term: -(√3/2)[(±(-√2)/3) + (√3)/6] = -(√3/2)*(±(-√2)/3) - (√3/2)*(√3)/6.Compute each part:First part: -(√3/2)*(±(-√2)/3) = ±(√3*√2)/(6) = ±(√6)/6.Second part: -(√3/2)*(√3)/6 = -(3)/12 = -1/4.So, combining the second term:±(√6)/6 - 1/4.Now, combining both terms:First term: (-1/12) - (±√6)/6.Second term: ±(√6)/6 - 1/4.Adding them together:(-1/12 - 1/4) + (- (±√6)/6 + ±√6)/6.Simplify:-1/12 - 3/12 = -4/12 = -1/3.For the √6 terms:- (±√6)/6 + ±√6)/6 = 0.Because the ± signs cancel out.Therefore, cos(2α + π/3) = -1/3.So, regardless of the sign of cosθ, the √6 terms cancel out, and we're left with -1/3.Therefore, the answer is -1/3, which is option B.This confirms the earlier result I obtained using the first method. So, I'm confident that the answer is indeed -1/3.**Final Answer**boxed{B}