💡Okay, so I have this problem where I need to find the area enclosed by three curves: y equals the square root of x, y equals 2 minus x, and y equals negative one-third of x. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, I think I need to visualize these curves to understand how they intersect and form a closed figure. Maybe drawing a rough sketch would help. Let me recall what each of these curves looks like.The curve y = sqrt(x) is a parabola that opens to the right, starting at the origin (0,0) and increasing slowly as x increases. The curve y = 2 - x is a straight line with a slope of -1 and a y-intercept at (0,2). So it starts at (0,2) and decreases as x increases. The third curve, y = -1/3 x, is also a straight line but with a gentler slope of -1/3, passing through the origin (0,0).Alright, so I can imagine these three curves intersecting each other at certain points, forming a sort of triangular or polygonal shape. To find the area enclosed by them, I need to determine the points where these curves intersect each other because those points will be the vertices of the enclosed figure.Let me start by finding the intersection points.First, let's find where y = sqrt(x) intersects with y = 2 - x. To do this, I can set the two equations equal to each other:sqrt(x) = 2 - xHmm, solving this equation might require squaring both sides to eliminate the square root. Let me try that.Squaring both sides:(sqrt(x))^2 = (2 - x)^2x = 4 - 4x + x^2Now, bringing all terms to one side:x^2 - 5x + 4 = 0This is a quadratic equation. Let me factor it:(x - 1)(x - 4) = 0So, the solutions are x = 1 and x = 4.Wait, but when x = 4, let's check if it satisfies the original equation:sqrt(4) = 2, and 2 - 4 = -2. So, 2 ≠ -2. That means x = 4 is an extraneous solution introduced by squaring both sides. So, the only valid intersection point is at x = 1.Plugging x = 1 back into y = sqrt(x), we get y = 1. So, the first intersection point is (1, 1).Next, let's find where y = sqrt(x) intersects with y = -1/3 x.Setting them equal:sqrt(x) = -1/3 xHmm, this seems a bit tricky because sqrt(x) is always non-negative, while -1/3 x is negative for positive x. The only point where they can intersect is at x = 0, since that's the only point where both sides are zero.So, plugging x = 0 into either equation, we get y = 0. So, the second intersection point is (0, 0).Now, let's find where y = 2 - x intersects with y = -1/3 x.Setting them equal:2 - x = -1/3 xLet me solve for x:2 = x - (1/3)x2 = (2/3)xMultiply both sides by 3/2:x = 3Plugging x = 3 back into y = -1/3 x, we get y = -1. So, the third intersection point is (3, -1).Alright, so now I have the three intersection points: (0, 0), (1, 1), and (3, -1). These are the vertices of the figure enclosed by the three curves.Now, to find the area S of this figure, I think I can use integration. Since the figure is bounded by these curves, I need to set up integrals that cover the area between the curves from one intersection point to another.Let me first sketch the rough graph in my mind. Starting from (0, 0), the curve y = sqrt(x) goes up to (1, 1), while y = -1/3 x goes down to (3, -1). The line y = 2 - x connects (0, 2) to (3, -1). So, the enclosed figure is a sort of triangle with vertices at (0, 0), (1, 1), and (3, -1).But wait, actually, the figure is not a triangle because the sides are curves, not straight lines. So, I need to be careful about how I set up the integrals.Let me think about dividing the area into two parts: from x = 0 to x = 1, and from x = 1 to x = 3.From x = 0 to x = 1, the upper boundary is y = sqrt(x), and the lower boundary is y = -1/3 x. So, the area between these two curves from 0 to 1 can be found by integrating (sqrt(x) - (-1/3 x)) dx from 0 to 1.From x = 1 to x = 3, the upper boundary is y = 2 - x, and the lower boundary is y = -1/3 x. So, the area between these two curves from 1 to 3 can be found by integrating (2 - x - (-1/3 x)) dx from 1 to 3.Wait, let me verify that. At x = 1, y = sqrt(1) = 1, and y = 2 - 1 = 1, so both curves meet at (1,1). Then, from x = 1 to x = 3, y = 2 - x is above y = -1/3 x because at x = 3, y = 2 - 3 = -1, which is the same as y = -1/3 * 3 = -1. So, yes, between x = 1 and x = 3, y = 2 - x is above y = -1/3 x.Therefore, the total area S is the sum of these two integrals.Let me write that down:S = ∫ from 0 to 1 [sqrt(x) - (-1/3 x)] dx + ∫ from 1 to 3 [(2 - x) - (-1/3 x)] dxSimplifying the integrands:First integral: sqrt(x) + (1/3)xSecond integral: 2 - x + (1/3)x = 2 - (2/3)xSo, S = ∫₀¹ [sqrt(x) + (1/3)x] dx + ∫₁³ [2 - (2/3)x] dxNow, let's compute each integral separately.First integral: ∫ [sqrt(x) + (1/3)x] dx from 0 to 1The integral of sqrt(x) is (2/3)x^(3/2), and the integral of (1/3)x is (1/6)x².So, evaluating from 0 to 1:[(2/3)(1)^(3/2) + (1/6)(1)²] - [(2/3)(0)^(3/2) + (1/6)(0)²] = (2/3 + 1/6) - 0 = (4/6 + 1/6) = 5/6Second integral: ∫ [2 - (2/3)x] dx from 1 to 3The integral of 2 is 2x, and the integral of -(2/3)x is -(1/3)x².So, evaluating from 1 to 3:[2(3) - (1/3)(3)²] - [2(1) - (1/3)(1)²] = [6 - 3] - [2 - 1/3] = (3) - (5/3) = 3 - 5/3 = 4/3Therefore, the total area S is 5/6 + 4/3.To add these fractions, I need a common denominator. 5/6 + 4/3 = 5/6 + 8/6 = 13/6.So, the area S is 13/6.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First integral:∫₀¹ sqrt(x) dx = [ (2/3)x^(3/2) ] from 0 to 1 = 2/3 - 0 = 2/3∫₀¹ (1/3)x dx = [ (1/6)x² ] from 0 to 1 = 1/6 - 0 = 1/6Adding them: 2/3 + 1/6 = 5/6. That seems correct.Second integral:∫₁³ 2 dx = [2x] from 1 to 3 = 6 - 2 = 4∫₁³ -(2/3)x dx = [ -(1/3)x² ] from 1 to 3 = -(9/3) - (-(1/3)) = -3 + 1/3 = -8/3Wait, hold on, that doesn't seem right.Wait, let me recompute the second integral.∫ [2 - (2/3)x] dx from 1 to 3Integral of 2 is 2x, integral of -(2/3)x is -(1/3)x².So, evaluating at 3: 2*3 - (1/3)*(3)^2 = 6 - 3 = 3Evaluating at 1: 2*1 - (1/3)*(1)^2 = 2 - 1/3 = 5/3Subtracting: 3 - 5/3 = 4/3. Okay, that's correct.So, the second integral is indeed 4/3.Adding the two integrals: 5/6 + 4/3 = 5/6 + 8/6 = 13/6.So, S = 13/6.Hmm, that seems reasonable. Let me just visualize again to make sure I didn't miss any regions or count any areas twice.From x = 0 to x = 1, the area is between y = sqrt(x) and y = -1/3 x, which is correctly captured by the first integral.From x = 1 to x = 3, the area is between y = 2 - x and y = -1/3 x, which is correctly captured by the second integral.So, adding these two areas gives the total enclosed area.Therefore, I think my answer is correct.**Final Answer**The area ( S ) is boxed{dfrac{13}{6}}.

💡First, determine the number of silver stickers by doubling the number of gold stickers:2 * 50 = 100 silver stickers.Next, calculate the number of bronze stickers by subtracting 20 from the number of silver stickers:100 - 20 = 80 bronze stickers.Add up all the stickers:50 (gold) + 100 (silver) + 80 (bronze) = 230 stickers.Finally, divide the total number of stickers by the number of stickers each student will receive to find the number of students:230 / 46 = 5 students.

💡First, I need to determine the multiple of the number of chocolate squares that Mike ate, which is less than the number Jenny ate by 5.I know that Mike ate 20 chocolate squares and Jenny ate 65. According to the problem, Jenny ate 5 more than a certain multiple of Mike's chocolate squares.To find the multiple, I can set up the equation:Jenny's chocolate squares = Mike's chocolate squares × multiple + 5Plugging in the known values:65 = 20 × multiple + 5Next, I'll solve for the multiple. First, subtract 5 from both sides:65 - 5 = 20 × multiple60 = 20 × multipleThen, divide both sides by 20:60 / 20 = multiple3 = multipleTherefore, the multiple is 3.

💡First, I need to determine the total number of students, which is 10. The average score is given as 23 points, so the total score for all students is 23 multiplied by 10, which equals 230 points.Next, I'll calculate the total score based on the given distribution of scores. The score of 30 is achieved by 2 students, contributing 60 points. The score of 25 is achieved by x students, contributing 25x points. The score of 20 is achieved by y students, contributing 20y points. The score of 15 is achieved by 1 student, contributing 15 points.Adding these contributions together gives the equation: 60 + 25x + 20y + 15 = 230. Simplifying this equation, I get 25x + 20y = 155. Dividing the entire equation by 5 to simplify further, I obtain 5x + 4y = 31.Since the total number of students is 10 and we already have 2 students with a score of 30 and 1 student with a score of 15, the remaining students are x + y = 7.Now, I have two equations:1. 5x + 4y = 312. x + y = 7I'll solve this system of equations. From the second equation, I can express y as y = 7 - x. Substituting this into the first equation gives 5x + 4(7 - x) = 31. Simplifying this, I get 5x + 28 - 4x = 31, which simplifies to x + 28 = 31. Solving for x, I find x = 3. Substituting x = 3 back into y = 7 - x, I get y = 4.Now that I have the values of x and y, I can determine the median and mode. The scores in ascending order are: 15, 20, 20, 20, 20, 25, 25, 25, 30, 30. The median is the average of the 5th and 6th scores, which are both 20 and 25. Therefore, the median is (20 + 25) / 2 = 22.5.The mode is the score that appears most frequently, which is 20, appearing 4 times.

💡Okay, so I have this problem where there are spheres with radius 1 arranged in the shape of a square pyramid. The bottom layer is a square with n² spheres, the next layer up has (n-1)² spheres, and so on until the top layer, which is just one sphere. I need to figure out the height of this pyramid.First, let me visualize this. A square pyramid made of spheres... So, each layer is a square grid of spheres, and each subsequent layer has one less sphere along each side. That means the base is an n by n grid, the next layer is (n-1) by (n-1), and so on until the top, which is just a single sphere.Since each sphere has a radius of 1, the diameter of each sphere is 2. That might be important because the vertical distance between the centers of the spheres in adjacent layers could be related to the diameter.Wait, but how exactly are these spheres stacked? In a square pyramid, each layer is centered over the one below it, right? So, the centers of the spheres in the upper layer are positioned directly above the centers of the spheres in the lower layer? Or are they offset?Hmm, in a typical square packing, each sphere in the upper layer sits in the "pocket" formed by four spheres below it. So, the centers of the spheres in the upper layer are not directly above the centers of the spheres in the lower layer but are offset. This offset would create a vertical distance between the centers of the spheres in adjacent layers.I think this vertical distance can be calculated using the Pythagorean theorem. If I consider two spheres in adjacent layers, their centers are separated by some vertical distance, and the horizontal distance between their centers is half the diagonal of a square formed by four spheres in the lower layer.Wait, let me think. The centers of four spheres in the lower layer form a square with side length 2 (since each sphere has a radius of 1, the distance between centers is 2). The diagonal of this square is 2√2. So, half of that diagonal is √2. So, the horizontal distance between the center of a sphere in the upper layer and the center of a sphere in the lower layer is √2.Now, the vertical distance between the centers can be found using the Pythagorean theorem, since the line connecting the centers of two spheres in adjacent layers forms a right triangle with the horizontal distance and the vertical distance. The hypotenuse of this triangle is the distance between the centers of the two spheres, which is 2 (since each sphere has a radius of 1, the distance between centers is 2).So, if the horizontal distance is √2 and the hypotenuse is 2, then the vertical distance (let's call it h) can be found by:h² + (√2)² = 2²h² + 2 = 4h² = 2h = √2Wait, so the vertical distance between the centers of the spheres in adjacent layers is √2? That seems right because the centers are offset both horizontally and vertically.But wait, in the problem, we're dealing with layers of spheres. Each layer is a square grid, so the vertical distance between the layers would be the vertical distance between the centers of the spheres in adjacent layers, which we just found to be √2.But hold on, the problem is asking for the height of the pyramid. The height would be the total vertical distance from the base to the top. Since each layer adds a vertical distance of √2, and there are n layers, does that mean the total height is n times √2?Wait, no. Because the base layer is on the ground, so the center of the base layer is at a height of 1 (since the radius is 1). Then, each subsequent layer adds a vertical distance of √2. So, the total height would be 1 (for the base) plus (n-1) times √2.Wait, let me clarify. The base layer has its spheres resting on the ground, so the center of each sphere in the base layer is at height 1. Then, the next layer is placed on top of the base layer, with its centers at height 1 + √2. The layer after that would be at height 1 + 2√2, and so on, until the top layer, which is the nth layer, would be at height 1 + (n-1)√2.But the total height of the pyramid would be the height of the top sphere's center plus its radius, right? Because the top sphere's center is at height 1 + (n-1)√2, and the top of the sphere is at height 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Wait, but is that correct? Because the base is on the ground, so the bottom of the base spheres are at height 0, their centers at height 1, and the top of the base spheres are at height 2. Similarly, the top sphere's center is at height 1 + (n-1)√2, and its top is at height 1 + (n-1)√2 + 1 = 2 + (n-1)√2.But the problem is asking for the height of the pyramid, which I think refers to the total height from the ground to the top of the pyramid. So, that would be 2 + (n-1)√2.Wait, but let me double-check. If n=1, then the pyramid is just a single sphere, so the height should be 2 (diameter). Plugging n=1 into 2 + (1-1)√2 = 2 + 0 = 2, which is correct.If n=2, the pyramid has two layers: a base of 4 spheres and a top sphere. The base layer's centers are at height 1, the top sphere's center is at height 1 + √2, and the top of the pyramid is at height 1 + √2 + 1 = 2 + √2. So, the height is 2 + √2, which seems correct.Wait, but let me think about the vertical stacking again. Each layer is placed such that the centers are vertically √2 apart. So, starting from the base layer at height 1, the next layer is at height 1 + √2, then 1 + 2√2, and so on. The top layer is the nth layer, so its center is at height 1 + (n-1)√2, and the top of that sphere is at 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Therefore, the total height of the pyramid is 2 + (n-1)√2.But wait, let me think again. If n=2, the height is 2 + √2, which is approximately 3.414. But if I think about two layers, the base layer has spheres with centers at 1, and the top sphere is placed on top, so its center is at 1 + √2, and the top of the sphere is at 1 + √2 + 1 = 2 + √2. That seems correct.Similarly, for n=3, the height would be 2 + 2√2, which is about 4.828. That seems reasonable.Wait, but I'm a bit confused because I thought the vertical distance between layers is √2, but maybe it's not. Let me re-examine the calculation.When I considered two spheres in adjacent layers, their centers are separated by a distance of 2 (since each sphere has radius 1, the distance between centers is 2). The horizontal distance between their centers is √2, as the centers are offset by half the diagonal of a square with side length 2. So, the horizontal distance is √2, and the vertical distance is h, such that h² + (√2)² = 2².So, h² + 2 = 4, so h² = 2, so h = √2. Therefore, the vertical distance between the centers of spheres in adjacent layers is indeed √2.Therefore, each layer adds √2 to the height. Starting from the base layer at height 1, the next layer is at 1 + √2, then 1 + 2√2, etc., up to the nth layer at 1 + (n-1)√2. The top of the pyramid is the top of the nth sphere, which is at 1 + (n-1)√2 + 1 = 2 + (n-1)√2.So, the height of the pyramid is 2 + (n-1)√2.Wait, but let me think about the base. The base is on the ground, so the bottom of the spheres are at height 0, their centers at 1, and the top of the base layer is at 2. The next layer is placed on top of the base layer, so its centers are at 1 + √2, and its top is at 1 + √2 + 1 = 2 + √2. So, the total height is 2 + (n-1)√2.Yes, that seems consistent.But wait, let me check with n=1. If n=1, the pyramid is just a single sphere, so the height should be 2. Plugging n=1 into 2 + (1-1)√2 = 2 + 0 = 2. Correct.For n=2, height is 2 + √2 ≈ 3.414. For n=3, it's 2 + 2√2 ≈ 4.828, and so on. That seems reasonable.Alternatively, if I consider the vertical distance between layers as √2, and there are (n-1) such distances between the base and the top, plus the radius at the top and bottom, but wait, the base is already on the ground, so we only need to add the radius at the top.Wait, no. The base is on the ground, so the bottom of the base spheres are at 0, their centers at 1, and the top of the base is at 2. Then, each subsequent layer adds √2 to the height, but the top of the pyramid is the top of the top sphere, which is at 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Yes, that seems correct.So, the height of the pyramid is 2 + (n-1)√2.But wait, let me think again. Is the vertical distance between layers √2 or is it something else? Because in a square packing, the vertical distance between layers is indeed √2, but I want to make sure.In a square packing, each sphere in the upper layer is placed in the "pocket" formed by four spheres in the lower layer. The centers of these four spheres form a square with side length 2, so the diagonal is 2√2. The center of the upper sphere is directly above the center of this square, so the horizontal distance from the upper sphere's center to each lower sphere's center is √2 (half the diagonal). The vertical distance can be found using the Pythagorean theorem, as the distance between centers is 2 (since each sphere has radius 1). So, vertical distance h satisfies h² + (√2)² = 2², so h² + 2 = 4, h² = 2, h = √2. So yes, the vertical distance between layers is √2.Therefore, the total height is 2 + (n-1)√2.Wait, but let me think about the number of gaps. If there are n layers, there are (n-1) gaps between them. Each gap contributes √2 to the height. So, starting from the base layer at height 1, the next layer is at 1 + √2, then 1 + 2√2, etc., up to the nth layer at 1 + (n-1)√2. The top of the pyramid is the top of the nth sphere, which is at 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Yes, that makes sense.Alternatively, if I think about the height as the sum of the radii at the bottom and top plus the sum of the vertical distances between the layers. The bottom radius is 1, the top radius is 1, and between the layers, there are (n-1) gaps each of √2. So, total height = 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Yes, that's another way to see it.So, I think the height of the pyramid is 2 + (n-1)√2.Wait, but let me check with n=2 again. If n=2, the height should be 2 + √2. Let's see: base layer at height 1, top sphere's center at 1 + √2, top of the top sphere at 1 + √2 + 1 = 2 + √2. Correct.Similarly, for n=3, it's 2 + 2√2, which is about 4.828. That seems reasonable.Therefore, the height of the pyramid is 2 + (n-1)√2.But wait, the problem says "Compute the height of the pyramid." So, is it 2 + (n-1)√2? Or is there another way to express it?Alternatively, it can be written as 2(1 + (n-1)(√2/2)), but that's not simpler. So, I think 2 + (n-1)√2 is the simplest form.Wait, but let me think again. The vertical distance between layers is √2, and there are (n-1) such distances. So, the total height from the base to the top of the pyramid is 2 (for the base and top radii) plus (n-1)√2. So, yes, 2 + (n-1)√2.Alternatively, if we consider the centers of the layers, the base center is at 1, the top center is at 1 + (n-1)√2, and the total height is from 0 to 1 + (n-1)√2 + 1 = 2 + (n-1)√2.Yes, that seems consistent.So, I think the height of the pyramid is 2 + (n-1)√2.But wait, let me think about the initial arrangement again. If the base is on the ground, the bottom of the spheres are at 0, their centers at 1, and the top of the base layer is at 2. Then, each subsequent layer adds √2 to the height. So, the top of the pyramid is at 2 + (n-1)√2.Yes, that's correct.Therefore, the height of the pyramid is 2 + (n-1)√2.Wait, but let me check if there's a different interpretation. Maybe the height is measured from the base to the top of the pyramid, which would be the same as what I calculated. So, yes, 2 + (n-1)√2.Alternatively, if someone thinks the height is just the distance between the centers of the base and the top, which would be (n-1)√2, but that doesn't include the radii at the top and bottom. So, I think including the radii is correct because the height of the pyramid should be from the ground to the top of the top sphere.Therefore, the height is 2 + (n-1)√2.Wait, but let me see if there's a different approach. Maybe considering the height as the sum of the diameters of the spheres along the vertical axis. But in a square pyramid, the spheres are not stacked directly on top of each other, so the vertical distance isn't just the sum of diameters. Instead, it's the sum of the vertical distances between the centers of the layers, which we found to be √2 each.So, the total height is 2 (for the base and top radii) plus (n-1)√2.Yes, that seems correct.Therefore, I think the height of the pyramid is 2 + (n-1)√2.

💡Okay, so I have this complex number expression to solve: (1 - i)^2 - (4 + 2i)/(1 - 2i) - 4i^2014. Hmm, that looks a bit intimidating at first glance, but I think I can break it down step by step.First, I need to remember how to handle complex numbers. I know that i is the imaginary unit, where i squared is -1. So, let's tackle each part of the expression one by one.Starting with (1 - i)^2. I think I can expand this using the formula (a - b)^2 = a^2 - 2ab + b^2. So, applying that here:(1 - i)^2 = 1^2 - 2*1*i + i^2 = 1 - 2i + i^2.Wait, i^2 is -1, right? So, substituting that in:1 - 2i + (-1) = (1 - 1) - 2i = 0 - 2i = -2i.Okay, so the first part simplifies to -2i. That wasn't too bad.Next, I have the fraction (4 + 2i)/(1 - 2i). Dividing complex numbers can be tricky, but I remember that to simplify such expressions, I need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (1 - 2i) is (1 + 2i). So, let's do that:(4 + 2i)/(1 - 2i) * (1 + 2i)/(1 + 2i) = [(4 + 2i)(1 + 2i)] / [(1 - 2i)(1 + 2i)].Let me compute the numerator first:(4 + 2i)(1 + 2i) = 4*1 + 4*2i + 2i*1 + 2i*2i = 4 + 8i + 2i + 4i^2.Again, i^2 is -1, so 4i^2 is -4. So, combining like terms:4 + 8i + 2i - 4 = (4 - 4) + (8i + 2i) = 0 + 10i = 10i.Now, the denominator:(1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4i^2.Since i^2 is -1, this becomes 1 - 4*(-1) = 1 + 4 = 5.So, putting it all together, the fraction simplifies to 10i / 5 = 2i.Alright, so the second part simplifies to 2i.Now, the last term is -4i^2014. I need to figure out what i^2014 is. I remember that the powers of i cycle every four terms:i^1 = ii^2 = -1i^3 = -ii^4 = 1And then it repeats: i^5 = i, and so on.So, to find i^2014, I can divide 2014 by 4 and find the remainder. Let's do that:2014 divided by 4 is 503 with a remainder of 2 (since 4*503 = 2012, and 2014 - 2012 = 2).So, i^2014 = i^(4*503 + 2) = (i^4)^503 * i^2 = 1^503 * (-1) = -1.Therefore, -4i^2014 = -4*(-1) = 4.Okay, so the last term simplifies to 4.Now, putting it all together:First part: -2iSecond part: -2i (because it's minus the fraction, which was 2i)Third part: +4So, combining them:-2i - 2i + 4 = (-2i - 2i) + 4 = (-4i) + 4 = 4 - 4i.Wait, let me double-check that. So, the first term is -2i, then subtracting 2i gives -4i, and then adding 4 gives 4 - 4i. Yeah, that seems right.I don't think I made any mistakes in the calculations. Let me just recap:1. (1 - i)^2 = -2i2. (4 + 2i)/(1 - 2i) = 2i3. -4i^2014 = 4So, combining: -2i - 2i + 4 = 4 - 4i.Yeah, that looks correct. So, the final answer should be 4 - 4i.**Final Answer**The value of the complex number is boxed{4 - 4i}.