💡First, I need to determine the growth of the papaya tree each year based on the given information.In the first year, the tree grows 2 feet.For the second year, it grows 50% more than the first year. To calculate this, I add 50% of 2 feet to the original 2 feet, resulting in 3 feet of growth.In the third year, the tree grows 50% more than the second year. Adding 50% of 3 feet to the original 3 feet gives 4.5 feet of growth.During the fourth year, the tree grows twice as much as it did in the third year. Multiplying 4.5 feet by 2 results in 9 feet of growth.Finally, in the fifth year, the tree grows half as much as it did in the fourth year. Dividing 9 feet by 2 gives 4.5 feet of growth.To find the total height of the tree after five years, I sum the growth for each year: 2 feet + 3 feet + 4.5 feet + 9 feet + 4.5 feet, which equals 23 feet.

💡Okay, so I have this problem with two parts, I and II. Let me try to tackle them one by one.Starting with part (I). The problem says that in the rectangular coordinate system xOy, curve C₁ is given by the parametric equations x = cosα and y = sinα, where α is a parameter. Hmm, that sounds familiar. I think that's the unit circle because x² + y² = cos²α + sin²α = 1. So, C₁ is the unit circle centered at the origin.Then, they mention establishing a polar coordinate system with the origin O as the pole and the positive x-axis as the polar axis. Curve C₂ has the polar equation ρ = 2cosθ. I remember that in polar coordinates, ρ = 2cosθ is a circle as well. To find its rectangular equation, I can use the conversion formulas: x = ρcosθ and y = ρsinθ. Also, ρ² = x² + y².So, starting with ρ = 2cosθ, if I multiply both sides by ρ, I get ρ² = 2ρcosθ. That translates to x² + y² = 2x. Let me rearrange that: x² - 2x + y² = 0. Completing the square for the x terms: (x² - 2x + 1) + y² = 1, so (x - 1)² + y² = 1. Ah, so C₂ is a circle with center at (1, 0) and radius 1.Now, to find the intersection points of C₁ and C₂. C₁ is x² + y² = 1, and C₂ is (x - 1)² + y² = 1. Let me subtract the equation of C₁ from C₂ to eliminate y²:(x - 1)² + y² - (x² + y²) = 1 - 1 Expanding (x - 1)²: x² - 2x + 1 + y² - x² - y² = 0 Simplify: -2x + 1 = 0 So, -2x + 1 = 0 => x = 1/2.Now that I have x = 1/2, plug this back into C₁ to find y:(1/2)² + y² = 1 1/4 + y² = 1 y² = 3/4 So, y = ±√(3)/2.Therefore, the intersection points are (1/2, √3/2) and (1/2, -√3/2). That seems straightforward.Moving on to part (II). Point A has polar coordinates (4, π/3). I need to find the maximum area of triangle AOB where point B is on curve C₂.First, let me recall that the area of triangle AOB can be calculated using the formula:Area = (1/2) * |OA| * |OB| * sin(∠AOB)Where |OA| and |OB| are the lengths of vectors OA and OB, and ∠AOB is the angle between them.Given that point A is (4, π/3) in polar coordinates, converting that to rectangular coordinates might help. Using the conversion formulas:x = ρcosθ = 4cos(π/3) = 4*(1/2) = 2 y = ρsinθ = 4sin(π/3) = 4*(√3/2) = 2√3So, point A is (2, 2√3) in rectangular coordinates.Point B is on curve C₂, which we already found is the circle (x - 1)² + y² = 1. So, point B can be represented in polar coordinates as (2cosθ, θ) because the polar equation of C₂ is ρ = 2cosθ.Wait, actually, in polar coordinates, ρ = 2cosθ, so the coordinates of B are (2cosθ, θ). Therefore, in rectangular coordinates, B is (2cosθ*cosθ, 2cosθ*sinθ) = (2cos²θ, 2cosθ sinθ). Hmm, that might complicate things. Maybe it's better to keep it in polar coordinates for the area formula.So, point A is (4, π/3) and point B is (2cosθ, θ). The angle between OA and OB is |π/3 - θ|. So, the area S is:S = (1/2) * |OA| * |OB| * sin(|π/3 - θ|)Since |OA| is 4 and |OB| is 2cosθ (because ρ = 2cosθ for point B), then:S = (1/2) * 4 * 2cosθ * sin(|π/3 - θ|) Simplify: S = 4cosθ * sin(|π/3 - θ|)But since sin is an odd function, sin(|π/3 - θ|) = sin(π/3 - θ) if π/3 - θ is positive, otherwise sin(θ - π/3). But since we're dealing with absolute value, maybe it's better to express it without the absolute value by considering the sine function's properties.Alternatively, since the area is always positive, we can write:S = 4cosθ * sin(π/3 - θ)But let me think about the angle difference. The angle between OA and OB is |π/3 - θ|, so sin(|π/3 - θ|) is the same as sin(π/3 - θ) if θ < π/3, and sin(θ - π/3) if θ > π/3. However, since sine is positive in both the first and second quadrants, and the area is positive regardless, maybe I can just write sin(π/3 - θ) but take the absolute value into account.Wait, actually, the area formula uses the sine of the angle between them, which is always positive because it's the magnitude. So, perhaps I can write:S = (1/2) * |OA| * |OB| * sin(∠AOB) = (1/2) * 4 * 2cosθ * sin(|π/3 - θ|) = 4cosθ * sin(|π/3 - θ|)But to make it easier, maybe I can express this without the absolute value by considering θ in the appropriate range. Let me assume θ is between 0 and 2π, so |π/3 - θ| can be expressed as π/3 - θ if θ < π/3, and θ - π/3 otherwise. But that might complicate differentiation.Alternatively, perhaps I can use the sine of a difference identity:sin(π/3 - θ) = sinπ/3 cosθ - cosπ/3 sinθ = (√3/2)cosθ - (1/2)sinθBut since we have sin(|π/3 - θ|), it's the same as |sin(π/3 - θ)| because sine is positive in the first and second quadrants. Wait, no, because |π/3 - θ| is between 0 and π, so sin(|π/3 - θ|) is the same as sin(π/3 - θ) if π/3 - θ is positive, otherwise sin(θ - π/3). But since sine is positive in both cases, maybe I can just write sin(π/3 - θ) and consider θ in the range where π/3 - θ is positive, but I'm not sure.Alternatively, maybe I can express the area in terms of θ without the absolute value and then find the maximum.So, let's write S = 4cosθ * sin(π/3 - θ). Let's expand sin(π/3 - θ):sin(π/3 - θ) = sinπ/3 cosθ - cosπ/3 sinθ = (√3/2)cosθ - (1/2)sinθTherefore, S = 4cosθ * [(√3/2)cosθ - (1/2)sinθ] = 4cosθ*(√3/2 cosθ - 1/2 sinθ) = 4*(√3/2 cos²θ - 1/2 cosθ sinθ) = 2√3 cos²θ - 2 cosθ sinθNow, to find the maximum of S, we can take the derivative with respect to θ and set it to zero.Let me compute dS/dθ:dS/dθ = d/dθ [2√3 cos²θ - 2 cosθ sinθ] = 2√3 * 2 cosθ (-sinθ) - 2 [ -sinθ sinθ + cosθ cosθ ] Wait, let me compute term by term.First term: 2√3 cos²θ Derivative: 2√3 * 2 cosθ (-sinθ) = -4√3 cosθ sinθSecond term: -2 cosθ sinθ Derivative: -2 [ -sinθ sinθ + cosθ cosθ ] = -2 [ -sin²θ + cos²θ ] = -2 cos(2θ) because cos²θ - sin²θ = cos2θWait, let me check:Derivative of -2 cosθ sinθ is:Using product rule: -2 [ -sinθ sinθ + cosθ cosθ ] = -2 [ -sin²θ + cos²θ ] = -2 (cos²θ - sin²θ) = -2 cos2θSo, overall derivative:dS/dθ = -4√3 cosθ sinθ - 2 cos2θSet derivative equal to zero:-4√3 cosθ sinθ - 2 cos2θ = 0 Let me factor out -2:-2(2√3 cosθ sinθ + cos2θ) = 0 So, 2√3 cosθ sinθ + cos2θ = 0Recall that sin2θ = 2 sinθ cosθ, so 2√3 cosθ sinθ = √3 sin2θThus, equation becomes:√3 sin2θ + cos2θ = 0Let me write this as:√3 sin2θ + cos2θ = 0This resembles a linear combination of sine and cosine. I can write this as R sin(2θ + φ) = 0, where R is the amplitude and φ is the phase shift.Compute R: R = √( (√3)^2 + 1^2 ) = √(3 + 1) = 2Compute φ: tanφ = (coefficient of cos)/(coefficient of sin) = 1/√3 => φ = π/6So, √3 sin2θ + cos2θ = 2 sin(2θ + π/6) = 0Thus, 2 sin(2θ + π/6) = 0 => sin(2θ + π/6) = 0Solutions: 2θ + π/6 = nπ, n integerSo, 2θ = -π/6 + nπ => θ = -π/12 + nπ/2We need θ in [0, 2π). Let's find all solutions:For n = 0: θ = -π/12 + 0 = -π/12 (not in [0, 2π))For n = 1: θ = -π/12 + π/2 = ( -π/12 + 6π/12 ) = 5π/12For n = 2: θ = -π/12 + π = ( -π/12 + 12π/12 ) = 11π/12For n = 3: θ = -π/12 + 3π/2 = ( -π/12 + 18π/12 ) = 17π/12For n = 4: θ = -π/12 + 2π = ( -π/12 + 24π/12 ) = 23π/12So, critical points at θ = 5π/12, 11π/12, 17π/12, 23π/12Now, we need to evaluate S at these θ values to find which gives the maximum area.Let me compute S for each θ:First, θ = 5π/12:Compute cosθ and sinθ:cos(5π/12) = cos(75°) = (√6 - √2)/4 ≈ 0.2588 sin(5π/12) = sin(75°) = (√6 + √2)/4 ≈ 0.9659Compute S = 2√3 cos²θ - 2 cosθ sinθFirst, cos²θ = [(√6 - √2)/4]^2 = (6 + 2 - 2√12)/16 = (8 - 4√3)/16 = (2 - √3)/4So, 2√3 cos²θ = 2√3 * (2 - √3)/4 = (2√3)(2 - √3)/4 = (4√3 - 6)/4 = √3 - 1.5Next, cosθ sinθ = [(√6 - √2)/4] * [(√6 + √2)/4] = (6 - 2)/16 = 4/16 = 1/4So, -2 cosθ sinθ = -2*(1/4) = -0.5Thus, S = (√3 - 1.5) - 0.5 = √3 - 2 ≈ 1.732 - 2 ≈ -0.268But area can't be negative, so maybe I made a mistake in the sign. Wait, S was defined as 4cosθ sin(π/3 - θ), which could be negative depending on θ. But since we're taking the absolute value earlier, maybe I should consider the absolute value. Alternatively, perhaps I should have taken the absolute value of the area expression.Wait, in the initial setup, I wrote S = 4cosθ sin(|π/3 - θ|), which is always positive. So, maybe I should have considered the absolute value when expanding. Alternatively, perhaps I should have kept the absolute value in the expression.Alternatively, maybe I should have expressed S as |4cosθ sin(π/3 - θ)|, which would make it always positive. So, perhaps I should take the absolute value of the expression before differentiating.But for the sake of time, let me compute the absolute value of S at each θ.At θ = 5π/12:S = |√3 - 2| ≈ |1.732 - 2| ≈ 0.268At θ = 11π/12:Compute cos(11π/12) = cos(165°) = -(√6 + √2)/4 ≈ -0.9659 sin(11π/12) = sin(165°) = (√6 - √2)/4 ≈ 0.2588Compute S = 2√3 cos²θ - 2 cosθ sinθcos²θ = [(√6 + √2)/4]^2 = (6 + 2 + 2√12)/16 = (8 + 4√3)/16 = (2 + √3)/4So, 2√3 cos²θ = 2√3*(2 + √3)/4 = (2√3)(2 + √3)/4 = (4√3 + 6)/4 = √3 + 1.5cosθ sinθ = [-(√6 + √2)/4] * [(√6 - √2)/4] = [-(6 - 2)]/16 = -4/16 = -1/4So, -2 cosθ sinθ = -2*(-1/4) = 0.5Thus, S = (√3 + 1.5) + 0.5 = √3 + 2 ≈ 1.732 + 2 ≈ 3.732So, |S| = 3.732At θ = 17π/12:cos(17π/12) = cos(255°) = -(√6 + √2)/4 ≈ -0.9659 sin(17π/12) = sin(255°) = -(√6 - √2)/4 ≈ -0.2588Compute S = 2√3 cos²θ - 2 cosθ sinθcos²θ = same as before: (2 + √3)/4So, 2√3 cos²θ = √3 + 1.5cosθ sinθ = [-(√6 + √2)/4] * [-(√6 - √2)/4] = [ (√6 + √2)(√6 - √2) ] / 16 = (6 - 2)/16 = 4/16 = 1/4So, -2 cosθ sinθ = -2*(1/4) = -0.5Thus, S = (√3 + 1.5) - 0.5 = √3 + 1 ≈ 1.732 + 1 ≈ 2.732At θ = 23π/12:cos(23π/12) = cos(345°) = (√6 - √2)/4 ≈ 0.2588 sin(23π/12) = sin(345°) = -(√6 + √2)/4 ≈ -0.9659Compute S = 2√3 cos²θ - 2 cosθ sinθcos²θ = (2 - √3)/4So, 2√3 cos²θ = √3 - 1.5cosθ sinθ = [(√6 - √2)/4] * [-(√6 + √2)/4] = [-(6 - 2)]/16 = -4/16 = -1/4So, -2 cosθ sinθ = -2*(-1/4) = 0.5Thus, S = (√3 - 1.5) + 0.5 = √3 - 1 ≈ 1.732 - 1 ≈ 0.732So, the areas at the critical points are approximately:θ = 5π/12: ~0.268 θ = 11π/12: ~3.732 θ = 17π/12: ~2.732 θ = 23π/12: ~0.732So, the maximum area occurs at θ = 11π/12, with S ≈ 3.732. But let's express this exactly.From earlier, at θ = 11π/12, S = √3 + 2. So, the maximum area is 2 + √3.Wait, but earlier I thought S was 4cosθ sin(π/3 - θ). Let me verify:At θ = 11π/12, π/3 - θ = π/3 - 11π/12 = (4π/12 - 11π/12) = -7π/12. The sine of that is sin(-7π/12) = -sin(7π/12). But since we're taking the absolute value, it's |sin(7π/12)| = sin(7π/12). So, S = 4cos(11π/12) sin(7π/12). Let me compute that:cos(11π/12) = -cos(π/12) = -(√6 + √2)/4 sin(7π/12) = sin(π - 5π/12) = sin(5π/12) = (√6 + √2)/4So, S = 4 * [-(√6 + √2)/4] * [(√6 + √2)/4] = 4 * [ - ( (√6 + √2)^2 ) / 16 ] = 4 * [ - (6 + 2 + 2√12 ) / 16 ] = 4 * [ - (8 + 4√3 ) / 16 ] = 4 * [ - (2 + √3)/4 ] = - (2 + √3 )But since area is positive, we take the absolute value, so S = 2 + √3.Yes, that matches. So, the maximum area is 2 + √3.I think that's the answer. Let me just recap:For part (I), the intersection points are (1/2, √3/2) and (1/2, -√3/2).For part (II), the maximum area of triangle AOB is 2 + √3.

💡Okay, so I have this problem here: Given the sets A equals {-1, 0, a} and B equals {x | 1 < 2^x < 2}, and it says that if A intersect B is not empty, then I need to find the range of the real number a. Hmm, okay, let me try to figure this out step by step.First, let me understand what each set is. Set A is straightforward: it has three elements, which are -1, 0, and a. Now, a is a real number, and I need to find its range based on the condition given. Set B is defined as {x | 1 < 2^x < 2}. That notation means that B is the set of all real numbers x such that 2 raised to the power of x is greater than 1 and less than 2. So, I need to figure out what values of x satisfy that inequality.Let me rewrite the inequality to make it clearer: 1 < 2^x < 2. I know that 2^0 is 1, and 2^1 is 2. So, if I have 2^x between 1 and 2, that means x must be between 0 and 1 because the function 2^x is increasing. So, when x is 0, 2^0 is 1, and when x is 1, 2^1 is 2. Therefore, for 2^x to be between 1 and 2, x has to be between 0 and 1. So, set B is the interval (0, 1).Alright, so B is all real numbers greater than 0 and less than 1. Now, the problem states that A intersect B is not empty. That means there has to be at least one element that is common to both sets A and B.Let me look at set A again: {-1, 0, a}. The elements are -1, 0, and a. Now, I need to see if any of these elements are in set B, which is (0, 1). Looking at -1: Is -1 in B? Well, B is (0, 1), so -1 is definitely not in B because it's less than 0. Next, 0: Is 0 in B? Again, B is (0, 1), which is an open interval, so it doesn't include 0. So, 0 is not in B either.That leaves us with a. So, for A intersect B to be non-empty, a must be in B. That means a has to satisfy the condition that 0 < a < 1. Wait, let me make sure I'm not missing anything here. If a is in B, then a must be between 0 and 1. But what if a is equal to 0 or 1? Well, since B is an open interval, it doesn't include 0 or 1. So, a can't be exactly 0 or exactly 1. It has to be strictly greater than 0 and strictly less than 1.Is there any other possibility? Let me think. If a was equal to -1, then A would be {-1, 0, -1}, but that's just {-1, 0}, and neither of those are in B. So, a can't be -1 either. So, the only way for A and B to have a common element is if a is in (0, 1). Therefore, the range of the real number a is all real numbers greater than 0 and less than 1.Let me just recap to make sure I didn't skip any steps. 1. Set A is {-1, 0, a}.2. Set B is {x | 1 < 2^x < 2}, which simplifies to (0, 1).3. A intersect B is not empty, so there must be at least one common element.4. The elements of A are -1, 0, and a. Neither -1 nor 0 are in B.5. Therefore, a must be in B, meaning 0 < a < 1.Yes, that seems to cover all the bases. I don't think I missed anything here. So, the range of a is (0, 1).**Final Answer**The range of the real number ( a ) is boxed{(0, 1)}.

💡First, I need to calculate the stock price at the end of the first year after an 80% increase from the initial price of 120.Next, I'll determine the stock price at the end of the second year by applying a 30% decrease to the price obtained at the end of the first year.Finally, I'll calculate the stock price at the end of the third year by increasing the second year's price by 50%.By following these steps sequentially, I can find the final stock price after the three-year period.

💡Okay, so I have this geometry problem here. Let me try to visualize it first. There's a diagram with points A, B, C, and D. Angle ABC is 90 degrees, which means it's a right angle. At point A, there's a smaller angle labeled as 3x degrees. Then, at point B, next to A, there's another angle labeled as 4x degrees minus 10. I need to find the value of x.Hmm, let me think about this step by step. Since angle ABC is 90 degrees, that's the big angle at point B. But the problem mentions two smaller angles at points A and B. Wait, actually, it says the smaller angle at A is 3x degrees, and the angle at B next to A is 4x - 10 degrees. So, are these two angles adjacent to each other at point B?Looking at the Asymptote code, it seems like point D is somewhere in the middle, connected to B. So, maybe the diagram has two lines from B: one going to C, making the right angle, and another going to D, creating these smaller angles. So, angle ABD is 3x degrees, and angle DBC is 4x - 10 degrees. Since ABC is 90 degrees, the sum of angles ABD and DBC should be 90 degrees.So, if I add 3x and (4x - 10), that should equal 90. Let me write that equation down:3x + (4x - 10) = 90Now, let me simplify this equation. Combine like terms:3x + 4x - 10 = 90That's 7x - 10 = 90Now, I can solve for x. First, add 10 to both sides:7x - 10 + 10 = 90 + 10Which simplifies to:7x = 100Now, divide both sides by 7:x = 100 / 7Hmm, 100 divided by 7 is approximately 14.2857, but since the problem doesn't specify rounding, I should keep it as a fraction. So, x is 100/7 degrees.Wait, let me double-check my steps to make sure I didn't make a mistake. I added 3x and 4x to get 7x, which seems right. Then I subtracted 10 and set it equal to 90. Adding 10 to both sides gives 100, and dividing by 7 gives x as 100/7. Yeah, that seems correct.I think that's the answer. So, x equals 100 over 7.

💡First, I identify the given information: the boat's speed in still water is unknown, denoted as ( x ) km/hr, and the current's speed is 3 km/hr. The boat travels 6.75 km downstream in 15 minutes.Next, I convert the time from minutes to hours since speed is in km/hr. 15 minutes is ( frac{1}{4} ) of an hour.When traveling downstream, the effective speed of the boat is the sum of its speed in still water and the current's speed, which is ( x + 3 ) km/hr.Using the formula for distance, ( text{Distance} = text{Speed} times text{Time} ), I set up the equation:[ 6.75 = (x + 3) times frac{1}{4} ]To solve for ( x ), I multiply both sides by 4:[ 27 = x + 3 ]Finally, I subtract 3 from both sides to find the speed of the boat in still water:[ x = 24 ]