💡First, I need to determine the total number of pieces of gum that John, Cole, and Aubrey have together after sharing equally. Since each person gets 33 pieces of gum and there are 3 people, the total number of pieces is 33 multiplied by 3, which equals 99 pieces.Next, I know that John has 54 pieces of gum and Cole has 45 pieces of gum. To find out how many pieces Aubrey has, I subtract the number of pieces John and Cole have from the total.So, 99 pieces (total) minus 54 pieces (John) minus 45 pieces (Cole) equals 0 pieces. Therefore, Aubrey has 0 pieces of gum.

💡To determine the number of unit cubes with no paint on them in a 6x6x6 cube where each face has a 4x4 grid painted in the middle, we can follow these steps:1. **Calculate the total number of unit cubes in the 6x6x6 cube:** [ 6 times 6 times 6 = 216 ]2. **Determine the number of painted unit squares on each face:** Each face has a 4x4 grid painted in the middle, so: [ 4 times 4 = 16 text{ painted squares per face} ]3. **Calculate the total number of painted unit squares across all six faces:** [ 16 times 6 = 96 text{ painted squares} ]4. **Identify overlapping painted cubes:** - **Corners:** Each corner of the 4x4 grid is shared by three faces. There are 4 corners per face, but each corner is counted three times (once for each adjacent face). Therefore, the number of unique corner cubes is: [ 4 text{ unique corner cubes} ] - **Edges (excluding corners):** Each edge of the 4x4 grid (excluding corners) is shared by two faces. Each face has 4 edges with 3 cubes each, so: [ 4 times 3 = 12 text{ edge cubes per face} ] However, each edge cube is counted twice (once for each adjacent face), so the number of unique edge cubes is: [ 12 text{ unique edge cubes} ]5. **Calculate the total number of unique painted cubes:** - **Corners:** 4 unique cubes - **Edges (excluding corners):** 12 unique cubes - **Remaining middle cubes:** Each 4x4 grid has (16 - 12 - 4 = 0) remaining middle cubes, which are unique to each face. Therefore, the total number of unique painted cubes is: [ 4 + 12 = 16 text{ unique painted cubes per face} ] However, since each face contributes 4 new cubes to the total, for 6 faces: [ 16 times 6 = 96 text{ painted squares, but accounted as } 4 times 6 + 16 = 40 text{ unique cubes} ]6. **Count the number of unpainted cubes:** [ 216 - 40 = 176 ]**Final Answer:**There are (boxed{176}) unit cubes with no paint on them.

💡First, I need to understand what the problem is asking. It wants to find what percentage of ( x ) is equal to 60% of 30% of ( x ).To approach this, I'll start by calculating 30% of ( x ). Converting 30% to a decimal gives 0.30. So, 30% of ( x ) is:[0.30 times x = 0.30x]Next, I need to find 60% of this result. Converting 60% to a decimal gives 0.60. Therefore, 60% of 0.30x is:[0.60 times 0.30x = 0.18x]Now, I have that 60% of 30% of ( x ) equals 0.18x. To express this as a percentage of ( x ), I recognize that 0.18x is the same as 18% of ( x ).Thus, the percentage of ( x ) that is equal to 60% of 30% of ( x ) is 18%.

💡Okay, so I have this hexagon ABCDEF, and I need to find its perimeter. The diagram is provided with some Asymptote code, which I can kind of visualize. Let me try to understand the structure of the hexagon based on the Asymptote code.First, the points are defined using a cis function, which I think stands for cosine and sine, so it's converting polar coordinates to Cartesian coordinates. The points are labeled A, B, C, D, E, F, and they are connected in order to form the hexagon. There are also some diagonals drawn from A to C, A to D, and A to E.Looking at the Asymptote code, point A is at (0,0). Point B is at cis(1, -pi/2), which would be (0, -1) because cis(r, theta) is (r*cos(theta), r*sin(theta)). So, cos(-pi/2) is 0, and sin(-pi/2) is -1, so B is at (0, -1). Point C is at cis(sqrt(3), -pi/3). Let me calculate that. Cos(-pi/3) is 0.5, and sin(-pi/3) is -sqrt(3)/2. So, multiplying by sqrt(3), the coordinates would be (sqrt(3)*0.5, sqrt(3)*(-sqrt(3)/2)) which simplifies to (sqrt(3)/2, -3/2). Point D is at cis(2, -pi/3 + atan(1/sqrt(2))). Hmm, that's a bit more complex. Let me break it down. The angle here is -pi/3 plus arctangent of 1/sqrt(2). Let me compute that angle first. Arctangent of 1/sqrt(2) is approximately 35.264 degrees, so in radians, that's about 0.615 radians. So, the total angle is -pi/3 (which is -1.047 radians) plus 0.615 radians, which is approximately -0.432 radians.Now, cos(-0.432) is approximately 0.907, and sin(-0.432) is approximately -0.420. Multiplying by radius 2, point D would be approximately (1.814, -0.840).Point E is at cis(sqrt(8), -pi/3 + atan(1/sqrt(2)) + atan(1/sqrt(3))). Let's compute the angle here. We already have -pi/3 + atan(1/sqrt(2)) which is approximately -0.432 radians. Adding atan(1/sqrt(3)) which is approximately 0.523 radians, so total angle is approximately -0.432 + 0.523 = 0.091 radians.So, cos(0.091) is approximately 0.995, and sin(0.091) is approximately 0.091. Multiplying by sqrt(8) which is approximately 2.828, point E is approximately (2.814, 0.258).Point F is at cis(3, -pi/3 + atan(1/sqrt(2)) + atan(1/sqrt(3)) + atan(1/sqrt(4))). Let's compute the angle. We have -pi/3 + atan(1/sqrt(2)) + atan(1/sqrt(3)) which is approximately -0.432 + 0.615 + 0.523 = approximately 0.706 radians. Adding atan(1/sqrt(4)) which is atan(0.5) approximately 0.464 radians. So total angle is approximately 0.706 + 0.464 = 1.170 radians.So, cos(1.170) is approximately 0.391, and sin(1.170) is approximately 0.920. Multiplying by radius 3, point F is approximately (1.173, 2.760).Now, looking at the Asymptote code, the sides are labeled with lengths: AB is 1, BC is 2, CD is 2, DE is 2, EF is 3, and FA is... wait, FA isn't labeled. Hmm, but in the initial problem, the user mentioned the perimeter, so maybe FA is also given? Wait, looking back, the Asymptote code labels "1" between A and B, "2" between B and C, "2" between C and D, "2" between D and E, and "3" between E and F. So, the side FA isn't labeled, but in the initial problem, the user says "What is the perimeter of the hexagon ABCDEF in this diagram?" and in the Asymptote code, the sides are labeled as 1, 2, 2, 2, 3, but not FA. So, maybe FA is another length that we need to compute.Wait, but in the initial problem, the user also provided a solution, which says:- Using the Pythagorean theorem: AC² = AB² + BC² = 1 + 2² = 5 ⇒ AC = sqrt(5) AD² = AC² + CD² = 5 + 2² = 9 ⇒ AD = 3 AE² = AD² + DE² = 9 + 2² = 13 ⇒ AE = sqrt(13) AF² = AE² + EF² = 13 + 3² = 22 ⇒ AF = sqrt(22) Then, the perimeter is 1 + 2 + 2 + 2 + 3 + sqrt(22) = 10 + sqrt(22).So, according to this, AF is sqrt(22), which is the sixth side. So, the perimeter is the sum of all sides: AB + BC + CD + DE + EF + FA = 1 + 2 + 2 + 2 + 3 + sqrt(22) = 10 + sqrt(22).But wait, I want to make sure that this approach is correct. Let me think through it step by step.First, the hexagon is drawn with points A, B, C, D, E, F. The sides AB, BC, CD, DE, EF are given as 1, 2, 2, 2, 3 respectively. The side FA is not given, so we need to compute it.The user's solution uses the Pythagorean theorem multiple times, starting from AB and BC to find AC, then using AC and CD to find AD, then AD and DE to find AE, and finally AE and EF to find AF.Is this approach valid? Let me see.First, triangle ABC: AB = 1, BC = 2, and angle at B. Wait, but is triangle ABC a right triangle? Because if it is, then AC² = AB² + BC². But in the Asymptote code, point B is at (0, -1), point C is at (sqrt(3)/2, -3/2). So, the vector from B to C is (sqrt(3)/2 - 0, -3/2 - (-1)) = (sqrt(3)/2, -1/2). So, the length BC is sqrt( (sqrt(3)/2)^2 + (-1/2)^2 ) = sqrt( 3/4 + 1/4 ) = sqrt(1) = 1. Wait, but in the Asymptote code, BC is labeled as 2. Hmm, that seems contradictory.Wait, maybe I made a mistake in interpreting the coordinates. Let me recalculate the distance between B and C.Point B is at (0, -1). Point C is at (sqrt(3)/2, -3/2). So, the distance BC is sqrt( (sqrt(3)/2 - 0)^2 + (-3/2 - (-1))^2 ) = sqrt( (3/4) + ( (-3/2 + 1) )^2 ) = sqrt( 3/4 + (-1/2)^2 ) = sqrt( 3/4 + 1/4 ) = sqrt(1) = 1. But in the Asymptote code, BC is labeled as 2. That seems inconsistent.Wait, maybe the labels are not corresponding to the actual distances? Or perhaps the Asymptote code is scaled differently. Let me check the Asymptote code again.Looking at the Asymptote code:- Point a is (0,0)- Point b is cis(1, -pi/2) which is (0, -1)- Point c is cis(sqrt(3), -pi/3) which is (sqrt(3)/2, -3/2)- Point d is cis(2, -pi/3 + atan(1/sqrt(2))) which is approximately (1.814, -0.840)- Point e is cis(sqrt(8), -pi/3 + atan(1/sqrt(2)) + atan(1/sqrt(3))) which is approximately (2.814, 0.258)- Point f is cis(3, -pi/3 + atan(1/sqrt(2)) + atan(1/sqrt(3)) + atan(1/sqrt(4))) which is approximately (1.173, 2.760)So, the distances between consecutive points:AB: distance from (0,0) to (0,-1) is 1. Correct, labeled as 1.BC: distance from (0,-1) to (sqrt(3)/2, -3/2). As I calculated earlier, this is 1, but in the Asymptote code, it's labeled as 2. Hmm, that's a problem.Wait, maybe the labels are not the actual lengths but something else? Or perhaps the Asymptote code is just for drawing, and the labels are correct. Alternatively, maybe the coordinates are scaled.Wait, in the Asymptote code, point c is at cis(sqrt(3), -pi/3). So, the radius is sqrt(3), which would make the distance from A to C equal to sqrt(3). But in the initial solution, AC is sqrt(5). That's conflicting.Wait, perhaps the Asymptote code is not to scale, and the labels are the actual lengths. So, even though the coordinates might suggest different lengths, the labels are correct. So, AB is 1, BC is 2, CD is 2, DE is 2, EF is 3, and FA is sqrt(22). Therefore, the perimeter is 1 + 2 + 2 + 2 + 3 + sqrt(22) = 10 + sqrt(22).But I want to make sure that this approach is correct. Let me think about the structure of the hexagon.From the Asymptote code, the hexagon is drawn with points A, B, C, D, E, F, and diagonals from A to C, A to D, and A to E. So, it's a star-shaped hexagon with A as the center. Therefore, the sides AB, BC, CD, DE, EF, and FA are the edges of the hexagon.Given that, the user's solution uses the Pythagorean theorem multiple times, treating each segment as a right triangle with the previous segments. So, starting from AB = 1, BC = 2, forming a right triangle ABC, then AC = sqrt(5). Then, using AC and CD = 2, forming another right triangle ACD, so AD = 3. Then, using AD and DE = 2, forming another right triangle ADE, so AE = sqrt(13). Finally, using AE and EF = 3, forming another right triangle AEF, so AF = sqrt(22).But wait, is each of these triangles actually right-angled? Because if they are, then the Pythagorean theorem applies. But in reality, the hexagon is not necessarily composed of right angles at each step. So, this approach might be incorrect unless each of these segments forms a right angle with the previous one.Looking back at the Asymptote code, the angles for each point are given. Let me check if the angles between the segments are right angles.For example, the angle at point B: the segment AB is from A(0,0) to B(0,-1), which is straight down. Then, segment BC goes from B(0,-1) to C(sqrt(3)/2, -3/2). The direction from B to C is towards the right and slightly up. So, the angle at B is not a right angle. Therefore, triangle ABC is not a right triangle, so AC² ≠ AB² + BC². Therefore, the initial step in the user's solution is incorrect.Hmm, that's a problem. So, the user's solution assumes that each triangle is right-angled, but in reality, the hexagon is not composed of right angles at each vertex. Therefore, the approach is flawed.So, how can I correctly find the perimeter? Well, the perimeter is the sum of all the side lengths. We have AB = 1, BC = 2, CD = 2, DE = 2, EF = 3, and FA is unknown. So, if I can find FA, then I can sum them all up.But how do I find FA? Since the hexagon is drawn with diagonals from A to C, A to D, and A to E, perhaps we can use those to find FA.Wait, in the Asymptote code, point F is at cis(3, ...). So, the distance from A to F is 3, but in the user's solution, AF is sqrt(22). That's conflicting. Wait, maybe the Asymptote code is not to scale, and the labels are correct. So, AF is sqrt(22), but in the code, it's drawn with radius 3. So, perhaps the labels are correct, and the Asymptote code is just for drawing purposes, not to scale.Alternatively, maybe the user's solution is incorrect because it assumes right angles where there are none.Wait, let me think differently. Maybe the hexagon is constructed in such a way that each segment after AB is at a right angle to the previous one. So, AB is vertical, BC is at a right angle to AB, CD is at a right angle to BC, and so on. If that's the case, then the user's solution would be correct.But looking at the Asymptote code, the angles are not 90 degrees apart. For example, point B is at -pi/2, point C is at -pi/3, which is a difference of pi/6, not pi/2. So, the angle between AB and BC is not 90 degrees.Therefore, the user's solution is incorrect because it assumes right angles where there are none.So, how can I correctly find the perimeter? Maybe I need to compute the lengths of all sides using the coordinates provided in the Asymptote code.Let me list the coordinates again:- A: (0, 0)- B: (0, -1)- C: (sqrt(3)/2, -3/2)- D: approximately (1.814, -0.840)- E: approximately (2.814, 0.258)- F: approximately (1.173, 2.760)Now, let's compute the distances between consecutive points.AB: distance from A to B = sqrt( (0 - 0)^2 + (-1 - 0)^2 ) = sqrt(0 + 1) = 1. Correct.BC: distance from B to C = sqrt( (sqrt(3)/2 - 0)^2 + (-3/2 - (-1))^2 ) = sqrt( (3/4) + ( (-1/2) )^2 ) = sqrt( 3/4 + 1/4 ) = sqrt(1) = 1. But in the Asymptote code, it's labeled as 2. Hmm, that's conflicting.Wait, maybe the labels are not the actual lengths but something else. Or perhaps the Asymptote code is scaled differently. Let me check the labels in the Asymptote code:label("1", (a+b)/2, NW);label("2", (b+c)/2, SW);label("2", (c+d)/2, SW);label("2", (d+e)/2, NE);label("3", (e+f)/2, NE);So, the labels are placed at the midpoints of the sides, with lengths 1, 2, 2, 2, 3. So, the side AB is labeled 1, BC is labeled 2, CD is labeled 2, DE is labeled 2, EF is labeled 3, and FA is not labeled. So, the actual lengths of AB, BC, CD, DE, EF are 1, 2, 2, 2, 3 respectively. Therefore, the distance from A to B is 1, B to C is 2, etc.But when I computed the distance from B to C using the coordinates, it was 1, not 2. So, that suggests that the coordinates are not to scale. Therefore, the labels are correct, and the coordinates are just for drawing purposes, not to scale.Therefore, I cannot rely on the coordinates to compute the actual lengths. Instead, I need to use the given labels and the structure of the hexagon to find the perimeter.Given that, the perimeter is the sum of all sides: AB + BC + CD + DE + EF + FA.We know AB = 1, BC = 2, CD = 2, DE = 2, EF = 3. We need to find FA.To find FA, perhaps we can use the fact that the hexagon is drawn with diagonals from A to C, A to D, and A to E. So, maybe we can use the lengths of these diagonals to find FA.Wait, in the user's solution, they used the Pythagorean theorem multiple times to find AC, AD, AE, and AF. Let me see if that makes sense.Starting with AB = 1 and BC = 2, assuming triangle ABC is right-angled at B, then AC = sqrt(1² + 2²) = sqrt(5). Then, using AC = sqrt(5) and CD = 2, assuming triangle ACD is right-angled at C, then AD = sqrt( (sqrt(5))² + 2² ) = sqrt(5 + 4) = 3. Then, using AD = 3 and DE = 2, assuming triangle ADE is right-angled at D, then AE = sqrt(3² + 2²) = sqrt(13). Finally, using AE = sqrt(13) and EF = 3, assuming triangle AEF is right-angled at E, then AF = sqrt( (sqrt(13))² + 3² ) = sqrt(13 + 9) = sqrt(22).Therefore, FA = sqrt(22), and the perimeter is 1 + 2 + 2 + 2 + 3 + sqrt(22) = 10 + sqrt(22).But as I thought earlier, this approach assumes that each triangle is right-angled, which may not be the case. However, since the labels are given as 1, 2, 2, 2, 3, and the diagonals are drawn from A, perhaps the hexagon is constructed in such a way that each segment after AB is at a right angle to the previous one, making each triangle right-angled.Alternatively, perhaps the hexagon is a spiral where each turn is 90 degrees, making each segment perpendicular to the previous one. If that's the case, then the user's solution is correct.But without more information, it's hard to be certain. However, given that the user's solution is provided and it's based on the assumption of right angles, and the labels are given as 1, 2, 2, 2, 3, I think the intended approach is to assume that each segment is perpendicular to the previous one, allowing the use of the Pythagorean theorem step by step.Therefore, the perimeter would be 1 + 2 + 2 + 2 + 3 + sqrt(22) = 10 + sqrt(22).So, after considering the possible issues with the coordinates and the assumption of right angles, I think the user's solution is correct based on the given labels and the structure of the hexagon.

💡To determine how many one-fourths are in five-eighths, I need to divide five-eighths by one-fourth.First, I'll set up the division of the two fractions:[frac{5}{8} div frac{1}{4}]Next, I'll convert the division into multiplication by taking the reciprocal of the second fraction:[frac{5}{8} times frac{4}{1}]Now, I'll multiply the numerators and the denominators:[frac{5 times 4}{8 times 1} = frac{20}{8}]Finally, I'll simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:[frac{20 div 4}{8 div 4} = frac{5}{2}]This means that there are five-halves, or 2.5, one-fourths in five-eighths.

💡Okay, so I have this problem about trigonometric functions, and I need to solve two parts. Let me take it step by step.First, part (1): The terminal side passes through the point P(-1, 2). I need to find sin α cos α. Hmm, okay. I remember that for any point (x, y) on the terminal side of an angle α in standard position, the trigonometric functions can be defined using x, y, and r, where r is the distance from the origin to the point. So, r is sqrt(x² + y²).Let me calculate r first. For point P(-1, 2), x is -1 and y is 2. So, r = sqrt((-1)² + 2²) = sqrt(1 + 4) = sqrt(5). Got that.Now, sin α is y/r, which is 2/sqrt(5). Cos α is x/r, which is -1/sqrt(5). So, sin α is positive because y is positive, and cos α is negative because x is negative. That makes sense because the point is in the second quadrant, right? Since x is negative and y is positive.So, sin α is 2/sqrt(5) and cos α is -1/sqrt(5). Now, I need to find sin α cos α. Let me multiply them together: (2/sqrt(5)) * (-1/sqrt(5)) = -2/(sqrt(5)*sqrt(5)) = -2/5. That seems straightforward. So, the value is -2/5.Wait, let me double-check. Multiplying 2/sqrt(5) and -1/sqrt(5) gives -2/(5). Yep, that looks right. So, part (1) is done.Now, moving on to part (2): The terminal side of angle α is on the line y = -3x. I need to find tan α + 3/cos α. Hmm, okay. So, the terminal side is on the line y = -3x, which is a straight line passing through the origin with slope -3. That means the angle α has a tangent of -3 because tan α is y/x, which is -3x/x = -3. So, tan α is -3.But wait, the terminal side could be in either the second or fourth quadrant because the line y = -3x passes through both quadrants. In the second quadrant, x is negative and y is positive, while in the fourth quadrant, x is positive and y is negative. So, I need to consider both cases.Let me handle each case separately.Case 1: α is in the second quadrant. So, x is negative, y is positive. Since tan α = -3, which is y/x = -3. Let me represent x as some negative value, say x = -a, where a is positive. Then y would be -3x = -3*(-a) = 3a. So, y is positive as expected.Now, let's find cos α. Cos α is x/r, where r is sqrt(x² + y²). Let's compute r: sqrt((-a)² + (3a)²) = sqrt(a² + 9a²) = sqrt(10a²) = a*sqrt(10). So, r = a*sqrt(10).Therefore, cos α = x/r = (-a)/(a*sqrt(10)) = -1/sqrt(10). Similarly, sin α = y/r = (3a)/(a*sqrt(10)) = 3/sqrt(10).Now, I need to compute tan α + 3/cos α. We already know tan α is -3. Let's compute 3/cos α: 3 divided by (-1/sqrt(10)) is 3*(-sqrt(10)) = -3*sqrt(10). So, tan α + 3/cos α is -3 + (-3*sqrt(10)) = -3 - 3*sqrt(10).Case 2: α is in the fourth quadrant. So, x is positive, y is negative. Again, tan α = -3, which is y/x = -3. Let me represent x as some positive value, say x = a, where a is positive. Then y would be -3x = -3a. So, y is negative as expected.Now, let's find cos α. Cos α is x/r, where r is sqrt(x² + y²). Let's compute r: sqrt(a² + (-3a)²) = sqrt(a² + 9a²) = sqrt(10a²) = a*sqrt(10). So, r = a*sqrt(10).Therefore, cos α = x/r = a/(a*sqrt(10)) = 1/sqrt(10). Similarly, sin α = y/r = (-3a)/(a*sqrt(10)) = -3/sqrt(10).Now, I need to compute tan α + 3/cos α. Again, tan α is -3. Let's compute 3/cos α: 3 divided by (1/sqrt(10)) is 3*sqrt(10). So, tan α + 3/cos α is -3 + 3*sqrt(10).Wait, let me make sure I didn't make a mistake here. In the second quadrant, cos α is negative, so 3/cos α is negative, which gives -3*sqrt(10). Adding tan α, which is -3, gives -3 - 3*sqrt(10). In the fourth quadrant, cos α is positive, so 3/cos α is positive, giving 3*sqrt(10). Adding tan α, which is -3, gives -3 + 3*sqrt(10). That seems correct.So, depending on the quadrant, the value of tan α + 3/cos α is either -3 - 3*sqrt(10) or -3 + 3*sqrt(10).Let me just recap to make sure I didn't skip any steps or make any calculation errors.For part (1):- Point P(-1, 2)- r = sqrt(1 + 4) = sqrt(5)- sin α = 2/sqrt(5)- cos α = -1/sqrt(5)- sin α cos α = (2/sqrt(5))*(-1/sqrt(5)) = -2/5Looks good.For part (2):- Line y = -3x, so tan α = -3- Two cases: second and fourth quadrants- Second quadrant: x negative, y positive - x = -a, y = 3a - r = a*sqrt(10) - cos α = -1/sqrt(10) - 3/cos α = -3*sqrt(10) - tan α + 3/cos α = -3 - 3*sqrt(10)- Fourth quadrant: x positive, y negative - x = a, y = -3a - r = a*sqrt(10) - cos α = 1/sqrt(10) - 3/cos α = 3*sqrt(10) - tan α + 3/cos α = -3 + 3*sqrt(10)Everything checks out. I think I've got it right.**Final Answer**For part (1): boxed{-dfrac{2}{5}}.For part (2): In the second quadrant, boxed{-3 - 3sqrt{10}}; in the fourth quadrant, boxed{-3 + 3sqrt{10}}.