💡First, identify the number of years Tony spent on each of his undergraduate degrees. He spent 4 years on his first degree in science and 2 more degrees, each taking 4 years.Next, calculate the total time spent on undergraduate degrees by adding these periods together: 4 years + (2 × 4 years) = 12 years.Finally, subtract the total undergraduate time from the overall education period to determine the duration of his graduate degree in physics: 14 years - 12 years = 2 years.

💡Alright, so I have this problem where I need to find the polar coordinates of a point P that's given in Cartesian coordinates as (1, -√3). Hmm, okay, I remember that polar coordinates are represented as (ρ, θ), where ρ is the distance from the origin to the point, and θ is the angle made with the positive x-axis. First, I think I need to find ρ. I recall that ρ is calculated using the formula ρ = √(x² + y²). So, plugging in the values from the Cartesian coordinates, x is 1 and y is -√3. Let me compute that:ρ = √(1² + (-√3)²) = √(1 + 3) = √4 = 2.Okay, so ρ is 2. That seems straightforward. Now, onto finding θ. I know that θ can be found using the arctangent function, specifically θ = arctan(y/x). Here, y is -√3 and x is 1, so:θ = arctan(-√3 / 1) = arctan(-√3).Hmm, arctan(-√3) is a value I should recognize. I remember that tan(π/3) = √3, so tan(-π/3) = -√3. Therefore, θ = -π/3.But wait, angles in polar coordinates are typically given between 0 and 2π, right? So, if θ is negative, I need to add 2π to it to get it within the standard range. Let's do that:θ = -π/3 + 2π = (6π/3 - π/3) = 5π/3.So, θ is 5π/3. That makes sense because the point (1, -√3) is in the fourth quadrant, and 5π/3 is indeed in the fourth quadrant.Let me just double-check my calculations to make sure I didn't make any mistakes. Starting with ρ:ρ = √(1² + (-√3)²) = √(1 + 3) = √4 = 2. Yep, that's correct.For θ, I used arctan(y/x) = arctan(-√3/1) = arctan(-√3). Since tan(π/3) = √3, tan(-π/3) = -√3, so θ = -π/3. Adding 2π gives θ = 5π/3, which is in the fourth quadrant. That all checks out.I think I've got it. The polar coordinates should be (2, 5π/3). Just to visualize, the point is 2 units away from the origin, and the angle is 5π/3 radians, which is 300 degrees, placing it in the fourth quadrant. That matches with the Cartesian coordinates (1, -√3), which is also in the fourth quadrant.Yeah, I'm pretty confident now. I don't think I missed anything here.

💡Okay, so I have this problem where I need to find the locus of points X inside or on the sides of a trapezoid ABCD, where BC is parallel to AD. The condition is that the area of triangle XAB is equal to the area of triangle XCD. Hmm, let me try to visualize this.First, I remember that a trapezoid has two sides that are parallel, which in this case are BC and AD. So, ABCD is a trapezoid with BC || AD. Now, I need to find all points X such that the area of triangle XAB equals the area of triangle XCD.Let me draw a rough sketch in my mind. Let's label the trapezoid with A and D as the endpoints of one base, and B and C as the endpoints of the other base. So, AD is the longer base, and BC is the shorter one, but it could be the other way around too. Anyway, BC is parallel to AD.Now, point X is somewhere inside or on the trapezoid. The areas of triangles XAB and XCD must be equal. I need to find all such points X.I recall that the area of a triangle is (1/2)*base*height. So, for triangle XAB, the base is AB, and the height would be the perpendicular distance from X to AB. Similarly, for triangle XCD, the base is CD, and the height is the perpendicular distance from X to CD.Wait, but AB and CD are not necessarily parallel. Hmm, actually, in a trapezoid, only BC and AD are parallel. So, AB and CD are the non-parallel sides, which are called the legs of the trapezoid.So, the bases are BC and AD, and the legs are AB and CD. That makes sense.So, the area of triangle XAB is (1/2)*AB*height1, where height1 is the distance from X to AB. Similarly, the area of triangle XCD is (1/2)*CD*height2, where height2 is the distance from X to CD.Given that these areas are equal, we have:(1/2)*AB*height1 = (1/2)*CD*height2Simplifying, AB*height1 = CD*height2Hmm, so the product of the length of AB and the distance from X to AB is equal to the product of the length of CD and the distance from X to CD.But AB and CD are the legs of the trapezoid, which are not necessarily equal. So, unless AB equals CD, which would make it an isosceles trapezoid, the distances from X to AB and CD would have to adjust accordingly.Wait, but the problem doesn't specify that it's an isosceles trapezoid, so I can't assume that AB equals CD.Hmm, maybe I need another approach. Let me think about coordinates.Yes, maybe assigning coordinates to the trapezoid would help. Let me place the trapezoid on a coordinate system.Let me denote point A at (0, 0), D at (d, 0), B at (a, h), and C at (b, h), since BC is parallel to AD, which is along the x-axis. So, BC is the top base, and AD is the bottom base.So, coordinates:- A: (0, 0)- D: (d, 0)- B: (a, h)- C: (b, h)Now, point X is somewhere inside or on the trapezoid. Let me denote X as (x, y).Now, I need to find the areas of triangles XAB and XCD.First, triangle XAB has vertices at X(x, y), A(0, 0), and B(a, h). The area can be calculated using the shoelace formula.Similarly, triangle XCD has vertices at X(x, y), C(b, h), and D(d, 0). I'll use the shoelace formula for this as well.Let me compute the area of triangle XAB.Using the shoelace formula:Area = (1/2)| (x*(0 - h) + 0*(h - y) + a*(y - 0) ) |= (1/2)| -xh + 0 + ay |= (1/2)| ay - xh |Similarly, the area of triangle XCD:Area = (1/2)| x*(h - 0) + b*(0 - y) + d*(y - h) |= (1/2)| xh - by + dy - dh |= (1/2)| xh + (d - b)y - dh |So, according to the problem, these two areas are equal:(1/2)| ay - xh | = (1/2)| xh + (d - b)y - dh |Multiplying both sides by 2:| ay - xh | = | xh + (d - b)y - dh |Hmm, absolute values complicate things, but maybe I can square both sides to eliminate them. Alternatively, consider the cases where the expressions inside the absolute values are positive or negative.But before that, maybe I can express this without absolute values by considering the orientation.Wait, since X is inside the trapezoid, the y-coordinate is between 0 and h. So, let's see:For triangle XAB, the area is (1/2)| ay - xh |. Since X is inside the trapezoid, the value ay - xh could be positive or negative depending on the position of X.Similarly, for triangle XCD, the area is (1/2)| xh + (d - b)y - dh |. Again, the expression inside could be positive or negative.But maybe instead of dealing with absolute values, I can consider the signed areas, keeping in mind the orientation.Alternatively, perhaps I can find a relationship between x and y without absolute values by assuming the expressions inside are positive or negative.Wait, maybe a better approach is to consider the ratio of the areas.But let me think differently. Maybe instead of coordinates, I can use vector methods or properties of trapezoids.Wait, another idea: in a trapezoid, the line joining the midpoints of the legs (AB and CD) is called the midline, and its length is the average of the lengths of the two bases. Moreover, this midline is parallel to both bases.But in this problem, we are dealing with areas of triangles, not midlines. Hmm.Wait, but maybe the locus is related to the midline. Because if X is on the midline, then perhaps the areas of the triangles XAB and XCD are equal.Let me test this idea.Suppose X is on the midline. Then, the distance from X to AB and the distance from X to CD would be related.Wait, but in a trapezoid, the midline is equidistant from both bases. So, the distance from X to AB would be equal to the distance from X to CD.But in our case, the areas of the triangles depend on both the base lengths and the heights.Wait, but AB and CD are the legs, not the bases. The bases are BC and AD.Wait, hold on, maybe I got confused earlier.In the trapezoid, the two parallel sides are called the bases. So, in this case, BC and AD are the bases, and AB and CD are the legs.So, the midline would connect the midpoints of AB and CD, and its length would be the average of the lengths of BC and AD.But in our problem, we are dealing with triangles XAB and XCD, which have bases AB and CD, respectively.Hmm, perhaps the midline is not directly related here.Wait, but maybe the set of points X where the areas of XAB and XCD are equal is another line.Let me think again about the coordinate system.We have:Area of XAB = (1/2)| ay - xh |Area of XCD = (1/2)| xh + (d - b)y - dh |Setting them equal:| ay - xh | = | xh + (d - b)y - dh |Let me drop the absolute values for a moment and consider the equation:ay - xh = xh + (d - b)y - dhSimplify:ay - xh = xh + (d - b)y - dhBring all terms to one side:ay - xh - xh - (d - b)y + dh = 0Combine like terms:ay - (d - b)y - 2xh + dh = 0Factor y terms:[ a - (d - b) ] y - 2xh + dh = 0Simplify inside the brackets:a - d + bSo:(a + b - d) y - 2xh + dh = 0Let me write this as:-2h x + (a + b - d) y + dh = 0Hmm, that's a linear equation in x and y, so it represents a straight line.Alternatively, if I consider the other case where the expressions inside the absolute values have opposite signs:ay - xh = - (xh + (d - b)y - dh )So:ay - xh = -xh - (d - b)y + dhSimplify:ay - xh + xh + (d - b)y - dh = 0Combine like terms:ay + (d - b)y - dh = 0Factor y:[ a + d - b ] y - dh = 0So:( a + d - b ) y = dhThus:y = (dh)/(a + d - b )Hmm, that's a horizontal line, but only if a + d - b ≠ 0.But in a trapezoid, the coordinates are such that a < b < d, assuming it's a convex trapezoid.Wait, actually, in my coordinate system, I placed A at (0,0), D at (d,0), B at (a,h), and C at (b,h). So, for it to be a trapezoid, the sides AB and CD must not be parallel, and BC must be parallel to AD.So, the order of the points is A, B, C, D, such that AB and CD are the legs, and BC and AD are the bases.Therefore, in terms of coordinates, a and b must satisfy 0 < a < b < d, right? Because otherwise, the sides would cross each other.So, a < b < d.Therefore, a + d - b is positive because a + d > b since a > 0 and d > b.So, y = (dh)/(a + d - b ) is a horizontal line somewhere inside the trapezoid.But wait, if I consider this, then the locus would consist of two lines: one is the line from the first case, and the other is this horizontal line.But that seems contradictory because the problem states that X lies within the trapezoid or on its sides, so the locus should be a single line or something else.Wait, maybe I made a mistake in assuming both cases. Let me think.When I set | ay - xh | = | xh + (d - b)y - dh |, this equation can represent two cases:1. ay - xh = xh + (d - b)y - dh2. ay - xh = - (xh + (d - b)y - dh )So, both cases need to be considered.From case 1, I got the equation:-2h x + (a + b - d) y + dh = 0From case 2, I got:y = (dh)/(a + d - b )So, the locus consists of two lines: one is the line from case 1, and the other is the horizontal line from case 2.But wait, does this make sense geometrically?If X is on the horizontal line y = (dh)/(a + d - b ), then the areas of triangles XAB and XCD are equal.Similarly, if X is on the line from case 1, the areas are equal.So, the locus is the union of these two lines.But in the trapezoid, these two lines might intersect or not.Wait, let me check if these two lines intersect.From case 1: -2h x + (a + b - d) y + dh = 0From case 2: y = (dh)/(a + d - b )Substitute y from case 2 into case 1:-2h x + (a + b - d)*(dh)/(a + d - b ) + dh = 0Simplify:-2h x + [ (a + b - d)(dh) ] / (a + d - b ) + dh = 0Note that (a + b - d) = -(d - a - b )So, [ (a + b - d)(dh) ] / (a + d - b ) = [ - (d - a - b ) dh ] / (a + d - b ) = - dhTherefore, substituting back:-2h x - dh + dh = 0 => -2h x = 0 => x = 0So, the two lines intersect at x=0, y = (dh)/(a + d - b )But in the trapezoid, x=0 is point A, so the intersection is at point A.Therefore, the two lines intersect at point A.Similarly, if I check for another intersection, but since it's a trapezoid, maybe they only intersect at A.Wait, but in the trapezoid, the line from case 1 might also pass through another point.Let me see.From case 1: -2h x + (a + b - d) y + dh = 0Let me rearrange it:( a + b - d ) y = 2h x - dhSo, y = [2h x - dh ] / ( a + b - d )Hmm, let me see if this line passes through any other vertices.At x=0, y = [ - dh ] / ( a + b - d ) = dh / ( d - a - b )But since a + b - d is negative (because a + b < d, as a < b < d), so d - a - b is positive.So, y = dh / ( d - a - b )Which is the same as y = (dh)/(a + d - b ) because a + d - b = d - b + a, which is the same as d - (b - a).Wait, actually, a + d - b is equal to d - (b - a). Since a < b, b - a is positive, so d - (b - a) is less than d.But in any case, y = (dh)/(a + d - b ) is the same as the horizontal line from case 2.Wait, no, in case 2, y is constant, but in case 1, y is a linear function of x.So, the line from case 1 passes through point A (0, y_A ), where y_A = dh / (a + d - b )But in the trapezoid, point A is at (0,0). So, unless y_A = 0, which would require dh = 0, but h is the height, so h ≠ 0, and d ≠ 0, so y_A ≠ 0.Therefore, the line from case 1 passes through (0, y_A ), which is not point A, but somewhere above it.Wait, but in our coordinate system, point A is at (0,0), so unless y_A = 0, which it isn't, the line from case 1 doesn't pass through A.Wait, but earlier, when I substituted case 2 into case 1, I found that they intersect at x=0, y = (dh)/(a + d - b )But in the trapezoid, x=0 is the side AD, but y can't be greater than h because the trapezoid's height is h.Wait, let me compute y = (dh)/(a + d - b )Since a + d - b is greater than d - b, and since a > 0, so a + d - b > d - b.But d - b is positive because d > b.So, y = (dh)/(a + d - b ) is less than h because a + d - b > d - b, so denominator is larger, hence y < h.So, this point is somewhere on the side AD, above point A.Similarly, the line from case 1 passes through this point and extends into the trapezoid.So, the locus is the union of two lines: one is the horizontal line y = (dh)/(a + d - b ), and the other is the line from case 1.But wait, in the trapezoid, the horizontal line y = (dh)/(a + d - b ) would intersect the sides AB and CD at some points.Similarly, the line from case 1 would intersect the sides as well.But I'm supposed to find all points X within or on the trapezoid where the areas of XAB and XCD are equal.So, the locus is the set of points lying on both lines within the trapezoid.But wait, actually, in the trapezoid, the two lines intersect only at one point, which is on side AD.Therefore, the locus is just the line segment from that intersection point to another intersection point on the opposite side.Wait, maybe not. Let me think.Alternatively, perhaps the locus is just the line segment connecting the midpoints of the two bases BC and AD.Wait, that's a common result in trapezoids. The line connecting the midpoints of the bases is called the midline, and it has some interesting properties.In fact, I think that the set of points where the areas of triangles XAB and XCD are equal is the midline.Wait, let me test this.Suppose X is the midpoint of BC. Then, the area of triangle XAB should be equal to the area of triangle XCD.Wait, let me compute.If X is the midpoint of BC, then its coordinates are ((a + b)/2, h )Compute area of XAB:Using shoelace formula:(1/2)| x_A(y_B - y_X) + x_B(y_X - y_A) + x_X(y_A - y_B) |Wait, maybe it's easier to compute using base and height.But since X is on BC, which is the top base, the distance from X to AB and CD would be the same as the height of the trapezoid.Wait, no, because AB and CD are the legs, not the bases.Wait, maybe I need to think differently.Alternatively, maybe using vectors.But perhaps it's better to go back to the coordinate system.Let me compute the area of triangle XAB when X is the midpoint of BC.Coordinates of X: ((a + b)/2, h )Compute area of XAB:Using shoelace formula:Area = (1/2)| x_A(y_B - y_X) + x_B(y_X - y_A) + x_X(y_A - y_B) |Plug in the values:= (1/2)| 0*(h - h) + a*(h - 0) + (a + b)/2*(0 - h) |= (1/2)| 0 + a*h + (a + b)/2*(-h) |= (1/2)| a h - (a + b)h / 2 |= (1/2)| (2a h - a h - b h ) / 2 |= (1/2)| (a h - b h ) / 2 |= (1/2)*( | (a - b) h | ) / 2= | (a - b) h | / 4Similarly, compute area of triangle XCD:Coordinates of X: ((a + b)/2, h )Using shoelace formula:Area = (1/2)| x_C(y_D - y_X) + x_D(y_X - y_C) + x_X(y_C - y_D) |= (1/2)| b*(0 - h) + d*(h - h) + (a + b)/2*(h - 0) |= (1/2)| -b h + 0 + (a + b)/2 * h |= (1/2)| -b h + (a + b) h / 2 |= (1/2)| (-2b h + a h + b h ) / 2 |= (1/2)| (a h - b h ) / 2 |= | (a - b) h | / 4So, indeed, when X is the midpoint of BC, the areas of XAB and XCD are equal.Similarly, if X is the midpoint of AD, which is at (d/2, 0 ), let's compute the areas.Area of XAB:Using shoelace formula:= (1/2)| 0*(h - 0) + a*(0 - 0) + d/2*(0 - h) |= (1/2)| 0 + 0 - d h / 2 |= (1/2)*(d h / 2 ) = d h / 4Area of XCD:= (1/2)| b*(0 - 0) + d*(0 - h) + d/2*(h - 0) |= (1/2)| 0 - d h + d h / 2 |= (1/2)| -d h / 2 | = d h / 4So, again, the areas are equal.Therefore, the midpoints of BC and AD lie on the locus.Moreover, the line connecting these midpoints is the midline of the trapezoid.Therefore, perhaps the entire midline is the locus.But wait, earlier, when I solved the equations, I got two lines: one horizontal and one slant.But in reality, the midline is a single line segment connecting the midpoints of BC and AD.So, maybe the locus is just this midline.But why did the equations give me two lines?Perhaps because when I set the absolute areas equal, it can correspond to two different lines, but within the trapezoid, only the midline is relevant.Alternatively, maybe the other line is outside the trapezoid or doesn't intersect it.Wait, in the coordinate system, the horizontal line y = (dh)/(a + d - b ) is inside the trapezoid because y < h.But does it lie entirely within the trapezoid?Let me see.Given that a < b < d, so a + d - b is greater than d - b, which is positive.So, (dh)/(a + d - b ) is less than h because a + d - b > d - b.Therefore, the horizontal line is somewhere between y=0 and y=h.But does this line intersect the trapezoid?Yes, it should intersect the legs AB and CD.Wait, but in our earlier analysis, the line from case 1 passes through (0, y_A ) where y_A = (dh)/(a + d - b ), which is on side AD, and extends into the trapezoid.But in reality, the midline is the line connecting the midpoints of BC and AD, which is different.Wait, perhaps I made a mistake in interpreting the equations.Alternatively, maybe the locus is indeed the midline.Wait, let me think about another point on the midline.Suppose X is the midpoint of the midline. Then, it's also the midpoint of the trapezoid.Compute areas of XAB and XCD.But without coordinates, it's hard to see.Alternatively, maybe I can use symmetry.If I reflect the trapezoid over its midline, then points above the midline are mapped to points below, and vice versa.Therefore, if X is on the midline, then the areas of XAB and XCD should be equal because of the symmetry.Wait, but in a general trapezoid, which is not necessarily symmetric, this reflection might not hold.Wait, unless it's an isosceles trapezoid.Wait, no, in any trapezoid, the midline is equidistant from both bases, but the legs may not be symmetric.Hmm, so maybe the areas of the triangles are equal only when X is on the midline.Wait, but earlier, when I took X as the midpoint of BC, which is on the midline, the areas were equal.Similarly, when X is the midpoint of AD, which is also on the midline, the areas were equal.Therefore, it's likely that the midline is the locus.But then, why did the equations give me another line?Wait, perhaps because the equations are considering both the midline and another line outside the trapezoid.But within the trapezoid, only the midline is relevant.Alternatively, maybe the other line is coinciding with the midline.Wait, let me check.From case 1, the equation was:-2h x + (a + b - d) y + dh = 0Let me see if the midline satisfies this equation.The midline connects the midpoints of BC and AD.Midpoint of BC: ((a + b)/2, h )Midpoint of AD: (d/2, 0 )So, the midline is the line segment connecting ((a + b)/2, h ) and (d/2, 0 )Let me find the equation of this line.First, compute the slope:m = (0 - h ) / (d/2 - (a + b)/2 ) = (-h ) / ( (d - a - b ) / 2 ) = (-2h ) / (d - a - b )So, the slope is m = (-2h ) / (d - a - b )Now, using point-slope form with midpoint of AD: (d/2, 0 )y - 0 = m (x - d/2 )So,y = [ (-2h ) / (d - a - b ) ] (x - d/2 )Let me rearrange this:y = [ (-2h ) / (d - a - b ) ] x + [ (-2h ) / (d - a - b ) ] (-d/2 )Simplify:y = [ (-2h ) / (d - a - b ) ] x + ( h d ) / (d - a - b )Multiply both sides by (d - a - b ):(d - a - b ) y = -2h x + h dRearranged:2h x + (d - a - b ) y - h d = 0Wait, comparing this with the equation from case 1:From case 1: -2h x + (a + b - d ) y + dh = 0Note that (a + b - d ) = -(d - a - b )So, case 1 equation can be written as:-2h x - (d - a - b ) y + dh = 0Which is:2h x + (d - a - b ) y - dh = 0Wait, that's exactly the same as the equation of the midline.So, the line from case 1 is the midline.Therefore, the equation from case 1 corresponds to the midline.From case 2, we had y = (dh)/(a + d - b )But since a + d - b = d - (b - a ) and d - a - b = -(a + b - d )Wait, let me compute:From case 2: y = (dh)/(a + d - b )But a + d - b = (d - b ) + aSince a < b < d, a + d - b is positive.But in the midline equation, we have:2h x + (d - a - b ) y - dh = 0If I solve for y:(d - a - b ) y = -2h x + dhSo,y = [ -2h x + dh ] / (d - a - b )Which is the same as:y = [ dh - 2h x ] / (d - a - b )So, if I set x=0, y = dh / (d - a - b ) which is the same as case 2.Therefore, the line from case 1 is the midline, and case 2 is just a specific point on the midline when x=0.Therefore, the locus is the midline, which is the line segment connecting the midpoints of BC and AD.Hence, the locus of points X is the line segment joining the midpoints of the bases BC and AD.**Final Answer**The locus of points ( X ) is the line segment connecting the midpoints of the bases ( BC ) and ( AD ). Thus, the final answer is (boxed{text{The line segment connecting the midpoints of the bases of the trapezoid}}).

💡Okay, so I need to find the original function ( f(x) ) given the transformations that were applied to it. Let me try to understand the problem step by step.First, the problem says that all the abscissas (which are the x-coordinates) of the points on the graph of ( y = f(x) ) are shortened to half of their original length. So, if I have a point ( (a, b) ) on the graph of ( f(x) ), after shortening the abscissas, the new point becomes ( left( frac{a}{2}, b right) ). This transformation affects the x-values, essentially compressing the graph horizontally by a factor of 2.Next, the resulting curve is shifted to the right by ( frac{pi}{3} ) units. Shifting a graph to the right by ( c ) units involves replacing ( x ) with ( x - c ) in the function. So, if after the first transformation the function is ( y = fleft( frac{x}{2} right) ), then shifting it to the right by ( frac{pi}{3} ) would give ( y = fleft( frac{x - frac{pi}{3}}{2} right) ).The final graph after these transformations is given as ( y = sinleft( x - frac{pi}{4} right) ). So, I can set up the equation:[fleft( frac{x - frac{pi}{3}}{2} right) = sinleft( x - frac{pi}{4} right)]My goal is to find ( f(x) ). To do this, I need to reverse the transformations. Let me work backwards.First, let's consider the shifting. The function ( fleft( frac{x - frac{pi}{3}}{2} right) ) is the result after shifting ( fleft( frac{x}{2} right) ) to the right by ( frac{pi}{3} ). To reverse this, I can shift it to the left by ( frac{pi}{3} ). Shifting to the left by ( frac{pi}{3} ) would replace ( x ) with ( x + frac{pi}{3} ), so:[fleft( frac{(x + frac{pi}{3}) - frac{pi}{3}}{2} right) = sinleft( (x + frac{pi}{3}) - frac{pi}{4} right)]Simplifying the left side:[fleft( frac{x}{2} right) = sinleft( x + frac{pi}{3} - frac{pi}{4} right)]Calculating ( frac{pi}{3} - frac{pi}{4} ):[frac{pi}{3} - frac{pi}{4} = frac{4pi - 3pi}{12} = frac{pi}{12}]So, the equation becomes:[fleft( frac{x}{2} right) = sinleft( x + frac{pi}{12} right)]Now, I need to reverse the shortening of the abscissas. The function ( fleft( frac{x}{2} right) ) is a horizontal compression of ( f(x) ) by a factor of 2. To reverse this compression, I need to horizontally stretch the function by a factor of 2. Stretching horizontally by a factor of 2 involves replacing ( x ) with ( 2x ) in the function. So, replacing ( x ) with ( 2x ) in ( fleft( frac{x}{2} right) ) gives:[f(x) = sinleft( 2x + frac{pi}{12} right)]Wait, that doesn't seem right. Let me check my steps again.Starting from:[fleft( frac{x}{2} right) = sinleft( x + frac{pi}{12} right)]If I let ( u = frac{x}{2} ), then ( x = 2u ). Substituting back into the equation:[f(u) = sinleft( 2u + frac{pi}{12} right)]So, replacing ( u ) with ( x ):[f(x) = sinleft( 2x + frac{pi}{12} right)]Hmm, that seems to be the case. But looking at the answer choices, option D is ( sin(2x + frac{pi}{12}) ), which matches this result. However, I need to make sure that I didn't make a mistake in reversing the transformations.Let me think again. The original function ( f(x) ) was horizontally compressed by a factor of 2, then shifted right by ( frac{pi}{3} ) to get ( sin(x - frac{pi}{4}) ). So, to reverse it, I first shift left by ( frac{pi}{3} ), then stretch horizontally by a factor of 2.Starting from ( sin(x - frac{pi}{4}) ), shifting left by ( frac{pi}{3} ) gives ( sinleft( x + frac{pi}{3} - frac{pi}{4} right) = sinleft( x + frac{pi}{12} right) ). Then, stretching horizontally by 2 gives ( sinleft( frac{x}{2} + frac{pi}{12} right) ). Wait, that contradicts my earlier result.I think I messed up the order of operations. Let me clarify:The transformations were applied in the order: first compress horizontally by 2, then shift right by ( frac{pi}{3} ). To reverse them, I need to first reverse the shift, then reverse the compression.So, starting from the final function ( sin(x - frac{pi}{4}) ), first reverse the shift by shifting left ( frac{pi}{3} ):[sinleft( x + frac{pi}{3} - frac{pi}{4} right) = sinleft( x + frac{pi}{12} right)]Then, reverse the horizontal compression by stretching horizontally by 2. Stretching horizontally by 2 replaces ( x ) with ( frac{x}{2} ):[sinleft( frac{x}{2} + frac{pi}{12} right)]So, ( f(x) = sinleft( frac{x}{2} + frac{pi}{12} right) ), which is option B.Wait, but earlier I thought it was D. I think I confused the order of operations. Let me double-check.When you compress horizontally by 2, it's equivalent to replacing ( x ) with ( 2x ). So, ( f(x) ) becomes ( f(2x) ). Then, shifting right by ( frac{pi}{3} ) replaces ( x ) with ( x - frac{pi}{3} ), so the function becomes ( f(2(x - frac{pi}{3})) = f(2x - frac{2pi}{3}) ).This equals ( sin(x - frac{pi}{4}) ). So,[f(2x - frac{2pi}{3}) = sin(x - frac{pi}{4})]To find ( f(x) ), let me set ( u = 2x - frac{2pi}{3} ), so ( x = frac{u + frac{2pi}{3}}{2} ).Substituting back:[f(u) = sinleft( frac{u + frac{2pi}{3}}{2} - frac{pi}{4} right)]Simplify the argument:[frac{u}{2} + frac{pi}{3} - frac{pi}{4} = frac{u}{2} + frac{4pi - 3pi}{12} = frac{u}{2} + frac{pi}{12}]So,[f(u) = sinleft( frac{u}{2} + frac{pi}{12} right)]Replacing ( u ) with ( x ):[f(x) = sinleft( frac{x}{2} + frac{pi}{12} right)]Yes, that's consistent with option B. So, my initial confusion was due to mixing up the order of transformations. It's important to remember that when reversing transformations, you need to reverse them in the opposite order they were applied.Therefore, the correct answer should be option B: ( sinleft( frac{x}{2} + frac{pi}{12} right) ).

💡First, I recognize that for a product of integers to be even, at least one of the integers must be even. This is because multiplying odd numbers together always results in an odd number, whereas introducing an even number makes the entire product even.Alice has chosen six positive integers in total. To maximize the number of odd integers she could have chosen, I need to minimize the number of even integers. Since only one even integer is required to make the product even, the remaining five integers can be odd.Therefore, the maximum number of odd integers Alice could have chosen is five.

💡First, I need to find the least number that satisfies the given conditions when divided by 34, 48, and 5, leaving remainders of 4, 6, and 2 respectively.To approach this, I'll adjust the remainders to find a common remainder for all three divisors. The remainders are 4, 6, and 2. By subtracting 2 from each remainder, I can align them to a common remainder of 2.Next, I'll find the least common multiple (LCM) of the adjusted divisors (34, 48, and 5). Breaking down each number into its prime factors:- 34 = 2 × 17- 48 = 2⁴ × 3- 5 = 5The LCM is the product of the highest powers of all prime factors present in the numbers:- LCM = 2⁴ × 3 × 5 × 17 = 16 × 3 × 5 × 17 = 4080Finally, I'll add the common remainder (2) to the LCM to find the least number:- Least number = LCM + 2 = 4080 + 2 = 4082Therefore, the least number that meets the specified conditions is 4082.