💡Alright, so I have this function f(x) defined piecewise. It's ln(x) when x is greater than 0, and 9^{-x} + 1 when x is less than or equal to 0. I need to find f(f(1)) + f(-log₃2). Hmm, okay, let's break this down step by step.First, let's figure out what f(1) is. Since 1 is greater than 0, I'll use the first part of the function, which is ln(x). So, f(1) = ln(1). I remember that the natural logarithm of 1 is 0 because e^0 = 1. So, f(1) = 0.Now, I need to find f(f(1)), which is f(0). Since 0 is less than or equal to 0, I'll use the second part of the function, which is 9^{-x} + 1. Plugging in 0 for x, I get 9^{-0} + 1. Any number to the power of 0 is 1, so this simplifies to 1 + 1, which is 2. So, f(f(1)) = 2.Next, I need to find f(-log₃2). First, let's understand what -log₃2 is. Log base 3 of 2 is a positive number because 2 is greater than 1 and the logarithm is defined for positive numbers. So, -log₃2 is a negative number, which means I'll use the second part of the function again, 9^{-x} + 1.But wait, let me make sure I'm interpreting this correctly. The function f(x) is defined as 9^{-x} + 1 when x ≤ 0. Since -log₃2 is negative, yes, I should use that part. So, f(-log₃2) = 9^{-(-log₃2)} + 1. Simplifying the exponent, the double negative becomes positive, so it's 9^{log₃2} + 1.Now, I need to compute 9^{log₃2}. I remember that 9 is 3 squared, so 9 = 3². Therefore, 9^{log₃2} can be rewritten as (3²)^{log₃2}. Using exponent rules, (a^b)^c = a^{b*c}, so this becomes 3^{2*log₃2}.Hmm, 2*log₃2 is the same as log₃2², which is log₃4. So, 3^{log₃4} is just 4 because 3 raised to the log base 3 of 4 is 4. Therefore, 9^{log₃2} = 4.So, f(-log₃2) = 4 + 1 = 5.Now, putting it all together, f(f(1)) + f(-log₃2) = 2 + 5 = 7.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting with f(1) = ln(1) = 0, that seems right. Then f(0) = 9^{0} + 1 = 1 + 1 = 2, that checks out. For f(-log₃2), I converted 9^{log₃2} correctly by expressing 9 as 3² and then applying exponent rules. That led me to 4, and adding 1 gives 5. Adding 2 and 5 gives 7, which is one of the answer choices.I think I did everything correctly. Maybe I should just verify the exponent part again. So, 9^{log₃2} = (3²)^{log₃2} = 3^{2*log₃2} = (3^{log₃2})² = 2² = 4. Yes, that still holds. So, I'm confident that the answer is 7.**Final Answer**boxed{D}

💡First, I need to determine how many roses Danielle had before any of them wilted. On the second day, she had 9 unwilted roses after half of them wilted. This means that before the second wilting, she had 18 roses.Next, I consider the first day. After half of the roses wilted overnight, she was left with 18 roses. Therefore, before the first wilting, she must have had 36 roses.Thus, Danielle had 36 roses after trading the box of chocolates.

💡Okay, so I have this problem here where I need to express a composite function as a single equation. The function is given in parts: y equals sine of u, u equals log of v, and v equals the square root of x. Hmm, let me try to break this down step by step.First, I know that a composite function means that one function is inside another. So in this case, y is a function of u, which is a function of v, which is a function of x. So ultimately, y is a function of x through these intermediate steps.Let me write down the given functions:1. y = sin(u)2. u = log(v)3. v = sqrt(x)Alright, so starting from the innermost function, v is the square root of x. Then, u is the logarithm of v, and finally, y is the sine of u. So, to express y as a single equation in terms of x, I need to substitute these functions step by step.First, let's substitute v into the equation for u. Since v is sqrt(x), then u becomes log(sqrt(x)). So now, u is expressed in terms of x.Next, I need to express y in terms of x by substituting u into the equation for y. So y becomes sin(log(sqrt(x))). That seems straightforward, but maybe I can simplify it further.I remember that the logarithm of a square root can be simplified using logarithm properties. Specifically, log(a^b) equals b times log(a). So, log(sqrt(x)) is the same as log(x^(1/2)), which equals (1/2) log(x). So, substituting that back into the equation for u, we have u = (1/2) log(x). Then, y = sin(u) becomes y = sin((1/2) log(x)).Let me double-check that to make sure I didn't make a mistake. Starting from v = sqrt(x), then u = log(v) = log(sqrt(x)) = (1/2) log(x). Then y = sin(u) = sin((1/2) log(x)). Yeah, that seems right.Is there any other way to write this? Maybe using natural logarithm instead of log, but unless specified, log usually refers to base 10, but in higher mathematics, it often refers to the natural logarithm. Hmm, the problem didn't specify, so I think it's safe to assume it's the natural logarithm, which is often denoted as ln. But since the problem uses log, I'll stick with log.So, putting it all together, the composite function y is equal to the sine of half the logarithm of x. That should be the single equation expressing y in terms of x through the given functions.I don't think I missed any steps. I substituted each function into the next one, simplified using logarithm properties, and arrived at the final expression. It seems pretty straightforward once I break it down step by step. I guess the key was recognizing that the logarithm of a square root can be simplified, which made the expression neater.Just to recap:1. Start with v = sqrt(x).2. Substitute v into u = log(v) to get u = log(sqrt(x)).3. Simplify log(sqrt(x)) to (1/2) log(x).4. Substitute u into y = sin(u) to get y = sin((1/2) log(x)).Yep, that covers all the steps. I think I'm confident with this solution.

💡First, I need to determine the total snack sales for every 6 movie tickets sold. This involves calculating the revenue from each type of snack separately.For the grain crackers, there are 3 packs sold at 2.25 each. Multiplying these gives the total revenue from grain crackers.Next, for the beverages, there are 4 bottles sold at 1.50 each. Multiplying these provides the total revenue from beverages.Similarly, for the chocolate bars, there are 4 bars sold at 1.00 each. Multiplying these gives the total revenue from chocolate bars.After calculating the revenue from each snack type, I will add them all together to find the total snack sales for 6 movie tickets.Finally, to find the average snack sales per movie ticket, I will divide the total snack sales by the number of movie tickets sold, which is 6. This will give me the approximate average amount of estimated snack sales per movie ticket sold.

💡First, I need to solve the equation (lg a - 2lg 2 = 1).I recall that (2lg 2) can be rewritten using the logarithm power rule as (lg 2^2 = lg 4).Substituting this back into the equation, it becomes (lg a - lg 4 = 1).Using the logarithm subtraction rule, this simplifies to (lg left(frac{a}{4}right) = 1).To eliminate the logarithm, I can rewrite this as (frac{a}{4} = 10^1 = 10).Finally, solving for (a) gives (a = 10 times 4 = 40).

💡Alright, so I have this problem about an ellipse, and I need to figure out its equation and then solve some related questions. Let me start by understanding the given information.First, the ellipse is given by the equation (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a > b > 0). The eccentricity (e) is (frac{sqrt{3}}{2}). I remember that the eccentricity of an ellipse is related to (a) and (c) (the distance from the center to each focus) by the formula (e = frac{c}{a}). So, if (e = frac{sqrt{3}}{2}), then (c = frac{sqrt{3}}{2}a).Also, the area of the rhombus formed by connecting the four vertices of the ellipse is 4. Hmm, the vertices of the ellipse are at ((pm a, 0)) and ((0, pm b)). So, connecting these four points would form a rhombus. The area of a rhombus can be calculated as (frac{1}{2} times d_1 times d_2), where (d_1) and (d_2) are the lengths of the diagonals. In this case, the diagonals are (2a) and (2b). So, the area is (frac{1}{2} times 2a times 2b = 2ab). The problem states that this area is 4, so (2ab = 4), which simplifies to (ab = 2).Now, I also know that for an ellipse, (c^2 = a^2 - b^2). Since (c = frac{sqrt{3}}{2}a), substituting this into the equation gives (left(frac{sqrt{3}}{2}aright)^2 = a^2 - b^2). Calculating the left side: (frac{3}{4}a^2 = a^2 - b^2). Rearranging, we get (b^2 = a^2 - frac{3}{4}a^2 = frac{1}{4}a^2), so (b = frac{a}{2}).We also have (ab = 2), and since (b = frac{a}{2}), substituting gives (a times frac{a}{2} = 2), which simplifies to (frac{a^2}{2} = 2), so (a^2 = 4), meaning (a = 2) (since (a > 0)). Then, (b = frac{a}{2} = 1).So, the equation of the ellipse is (frac{x^2}{4} + y^2 = 1).Alright, that was part (Ⅰ). Now, moving on to part (Ⅱ). We have a line (l) intersecting the ellipse at two distinct points (A) and (B), with (A) at ((-a, 0)), which is ((-2, 0)) based on our previous result.(i) The distance between (A) and (B) is given as (frac{4sqrt{2}}{5}). I need to find the slope angle of line (l).Let me denote the slope of line (l) as (k). Since the line passes through point (A(-2, 0)), its equation can be written as (y = k(x + 2)).To find the other intersection point (B), I need to solve the system of equations:1. (frac{x^2}{4} + y^2 = 1)2. (y = k(x + 2))Substituting equation 2 into equation 1:(frac{x^2}{4} + [k(x + 2)]^2 = 1)Expanding this:(frac{x^2}{4} + k^2(x^2 + 4x + 4) = 1)Multiply through by 4 to eliminate the denominator:(x^2 + 4k^2(x^2 + 4x + 4) = 4)Expanding further:(x^2 + 4k^2x^2 + 16k^2x + 16k^2 = 4)Combine like terms:((1 + 4k^2)x^2 + 16k^2x + (16k^2 - 4) = 0)This is a quadratic equation in (x). Since we know one of the solutions is (x = -2) (point (A)), we can factor this out or use Vieta's formulas. Let me use Vieta's formulas.For a quadratic equation (ax^2 + bx + c = 0), the sum of the roots is (-b/a) and the product is (c/a).Let the roots be (x_1 = -2) and (x_2). Then:Sum of roots: (-2 + x_2 = -frac{16k^2}{1 + 4k^2})Product of roots: (-2 times x_2 = frac{16k^2 - 4}{1 + 4k^2})From the sum:(x_2 = -frac{16k^2}{1 + 4k^2} + 2 = frac{-16k^2 + 2(1 + 4k^2)}{1 + 4k^2} = frac{-16k^2 + 2 + 8k^2}{1 + 4k^2} = frac{-8k^2 + 2}{1 + 4k^2})From the product:(-2x_2 = frac{16k^2 - 4}{1 + 4k^2})So,(x_2 = frac{4 - 16k^2}{2(1 + 4k^2)} = frac{2 - 8k^2}{1 + 4k^2})Wait, that's consistent with what I got from the sum. So, the x-coordinate of point (B) is (frac{2 - 8k^2}{1 + 4k^2}). Then, the y-coordinate is (y = k(x + 2)), so:(y = kleft(frac{2 - 8k^2}{1 + 4k^2} + 2right) = kleft(frac{2 - 8k^2 + 2(1 + 4k^2)}{1 + 4k^2}right) = kleft(frac{2 - 8k^2 + 2 + 8k^2}{1 + 4k^2}right) = kleft(frac{4}{1 + 4k^2}right) = frac{4k}{1 + 4k^2})So, point (B) is (left(frac{2 - 8k^2}{1 + 4k^2}, frac{4k}{1 + 4k^2}right)).Now, the distance between (A(-2, 0)) and (Bleft(frac{2 - 8k^2}{1 + 4k^2}, frac{4k}{1 + 4k^2}right)) is given as (frac{4sqrt{2}}{5}).Let me compute this distance.The distance formula is:(|AB| = sqrt{left(frac{2 - 8k^2}{1 + 4k^2} - (-2)right)^2 + left(frac{4k}{1 + 4k^2} - 0right)^2})Simplify the x-component:(frac{2 - 8k^2}{1 + 4k^2} + 2 = frac{2 - 8k^2 + 2(1 + 4k^2)}{1 + 4k^2} = frac{2 - 8k^2 + 2 + 8k^2}{1 + 4k^2} = frac{4}{1 + 4k^2})So, the x-component is (frac{4}{1 + 4k^2}), and the y-component is (frac{4k}{1 + 4k^2}).Therefore, the distance squared is:(left(frac{4}{1 + 4k^2}right)^2 + left(frac{4k}{1 + 4k^2}right)^2 = frac{16}{(1 + 4k^2)^2} + frac{16k^2}{(1 + 4k^2)^2} = frac{16(1 + k^2)}{(1 + 4k^2)^2})So, the distance is (sqrt{frac{16(1 + k^2)}{(1 + 4k^2)^2}} = frac{4sqrt{1 + k^2}}{1 + 4k^2})We are told that this distance is (frac{4sqrt{2}}{5}), so:(frac{4sqrt{1 + k^2}}{1 + 4k^2} = frac{4sqrt{2}}{5})Divide both sides by 4:(frac{sqrt{1 + k^2}}{1 + 4k^2} = frac{sqrt{2}}{5})Square both sides:(frac{1 + k^2}{(1 + 4k^2)^2} = frac{2}{25})Multiply both sides by (25(1 + 4k^2)^2):(25(1 + k^2) = 2(1 + 4k^2)^2)Expand the right side:(2(1 + 8k^2 + 16k^4) = 2 + 16k^2 + 32k^4)So, the equation becomes:(25 + 25k^2 = 2 + 16k^2 + 32k^4)Bring all terms to one side:(32k^4 + 16k^2 + 2 - 25 - 25k^2 = 0)Simplify:(32k^4 - 9k^2 - 23 = 0)Let me set (u = k^2), so the equation becomes:(32u^2 - 9u - 23 = 0)Use the quadratic formula:(u = frac{9 pm sqrt{81 + 4 times 32 times 23}}{2 times 32})Calculate the discriminant:(81 + 4 times 32 times 23 = 81 + 2944 = 3025)So,(u = frac{9 pm sqrt{3025}}{64} = frac{9 pm 55}{64})So, two solutions:1. (u = frac{9 + 55}{64} = frac{64}{64} = 1)2. (u = frac{9 - 55}{64} = frac{-46}{64} = -frac{23}{32})Since (u = k^2) cannot be negative, we discard the second solution. So, (k^2 = 1), which means (k = pm 1).Therefore, the slope of line (l) is 1 or -1. The slope angle is the angle that the line makes with the positive x-axis. Since the slope is 1, the angle is (frac{pi}{4}), and since the slope is -1, the angle is (frac{3pi}{4}).So, the slope angles are (frac{pi}{4}) and (frac{3pi}{4}).Now, moving on to part (ii). We have point (Q(0, y_0)) on the perpendicular bisector of segment (AB), and (overrightarrow{QA} cdot overrightarrow{QB} = 4). We need to find (y_0).First, let me recall that the perpendicular bisector of a segment passes through the midpoint of the segment and is perpendicular to it. So, I need to find the midpoint of (AB) and then find the equation of the perpendicular bisector.From part (i), we have point (A(-2, 0)) and point (Bleft(frac{2 - 8k^2}{1 + 4k^2}, frac{4k}{1 + 4k^2}right)). Let me denote (B) as ((x_1, y_1)) for simplicity.The midpoint (M) of (AB) is:(M_x = frac{-2 + x_1}{2})(M_y = frac{0 + y_1}{2})So,(M_x = frac{-2 + frac{2 - 8k^2}{1 + 4k^2}}{2} = frac{-2(1 + 4k^2) + 2 - 8k^2}{2(1 + 4k^2)} = frac{-2 - 8k^2 + 2 - 8k^2}{2(1 + 4k^2)} = frac{-16k^2}{2(1 + 4k^2)} = frac{-8k^2}{1 + 4k^2})(M_y = frac{0 + frac{4k}{1 + 4k^2}}{2} = frac{2k}{1 + 4k^2})So, midpoint (M) is (left(frac{-8k^2}{1 + 4k^2}, frac{2k}{1 + 4k^2}right)).The slope of segment (AB) is (k), so the slope of the perpendicular bisector is (-frac{1}{k}).Therefore, the equation of the perpendicular bisector is:(y - M_y = -frac{1}{k}(x - M_x))Substituting (M_x) and (M_y):(y - frac{2k}{1 + 4k^2} = -frac{1}{k}left(x - frac{-8k^2}{1 + 4k^2}right))Simplify the equation:(y - frac{2k}{1 + 4k^2} = -frac{1}{k}left(x + frac{8k^2}{1 + 4k^2}right))Multiply through by (k) to eliminate the denominator:(kleft(y - frac{2k}{1 + 4k^2}right) = -left(x + frac{8k^2}{1 + 4k^2}right))Expand:(ky - frac{2k^2}{1 + 4k^2} = -x - frac{8k^2}{1 + 4k^2})Bring all terms to one side:(ky + x - frac{2k^2}{1 + 4k^2} + frac{8k^2}{1 + 4k^2} = 0)Simplify the constants:(- frac{2k^2}{1 + 4k^2} + frac{8k^2}{1 + 4k^2} = frac{6k^2}{1 + 4k^2})So, the equation becomes:(ky + x + frac{6k^2}{1 + 4k^2} = 0)But point (Q(0, y_0)) lies on this perpendicular bisector, so substituting (x = 0) and (y = y_0):(k y_0 + 0 + frac{6k^2}{1 + 4k^2} = 0)Solve for (y_0):(k y_0 = -frac{6k^2}{1 + 4k^2})Assuming (k neq 0), divide both sides by (k):(y_0 = -frac{6k}{1 + 4k^2})So, (y_0 = -frac{6k}{1 + 4k^2}). That's one relationship.Now, we also have the condition (overrightarrow{QA} cdot overrightarrow{QB} = 4).First, let's find vectors (overrightarrow{QA}) and (overrightarrow{QB}).Point (Q) is ((0, y_0)), so:(overrightarrow{QA} = A - Q = (-2 - 0, 0 - y_0) = (-2, -y_0))(overrightarrow{QB} = B - Q = (x_1 - 0, y_1 - y_0) = (x_1, y_1 - y_0))So, the dot product is:(overrightarrow{QA} cdot overrightarrow{QB} = (-2)(x_1) + (-y_0)(y_1 - y_0) = -2x_1 - y_0 y_1 + y_0^2)We are told this equals 4:(-2x_1 - y_0 y_1 + y_0^2 = 4)From earlier, we have expressions for (x_1), (y_1), and (y_0) in terms of (k):(x_1 = frac{2 - 8k^2}{1 + 4k^2})(y_1 = frac{4k}{1 + 4k^2})(y_0 = -frac{6k}{1 + 4k^2})Let me substitute these into the equation:(-2 times frac{2 - 8k^2}{1 + 4k^2} - left(-frac{6k}{1 + 4k^2}right) times frac{4k}{1 + 4k^2} + left(-frac{6k}{1 + 4k^2}right)^2 = 4)Simplify each term:First term: (-2 times frac{2 - 8k^2}{1 + 4k^2} = frac{-4 + 16k^2}{1 + 4k^2})Second term: (- left(-frac{6k}{1 + 4k^2}right) times frac{4k}{1 + 4k^2} = frac{24k^2}{(1 + 4k^2)^2})Third term: (left(-frac{6k}{1 + 4k^2}right)^2 = frac{36k^2}{(1 + 4k^2)^2})So, putting it all together:(frac{-4 + 16k^2}{1 + 4k^2} + frac{24k^2}{(1 + 4k^2)^2} + frac{36k^2}{(1 + 4k^2)^2} = 4)Combine the second and third terms:(frac{24k^2 + 36k^2}{(1 + 4k^2)^2} = frac{60k^2}{(1 + 4k^2)^2})So, the equation becomes:(frac{-4 + 16k^2}{1 + 4k^2} + frac{60k^2}{(1 + 4k^2)^2} = 4)Let me write (frac{-4 + 16k^2}{1 + 4k^2}) as (frac{-4 + 16k^2}{1 + 4k^2} = frac{16k^2 - 4}{1 + 4k^2})So, the equation is:(frac{16k^2 - 4}{1 + 4k^2} + frac{60k^2}{(1 + 4k^2)^2} = 4)Let me denote (u = 1 + 4k^2) to simplify the equation.Then, (16k^2 - 4 = 4(4k^2 - 1) = 4(u - 1 - 1) = 4(u - 2)). Wait, actually, let me compute directly:(16k^2 - 4 = 4(4k^2 - 1) = 4(u - 1 - 1) = 4(u - 2)). Hmm, maybe not the best approach.Alternatively, express everything in terms of (u):(16k^2 - 4 = 4(4k^2 - 1) = 4(u - 1 - 1) = 4(u - 2)), but actually, (u = 1 + 4k^2), so (4k^2 = u - 1). Therefore, (16k^2 = 4(u - 1)), so (16k^2 - 4 = 4(u - 1) - 4 = 4u - 4 - 4 = 4u - 8).Similarly, (60k^2 = 15 times 4k^2 = 15(u - 1)).So, substituting into the equation:(frac{4u - 8}{u} + frac{15(u - 1)}{u^2} = 4)Simplify each term:First term: (frac{4u - 8}{u} = 4 - frac{8}{u})Second term: (frac{15(u - 1)}{u^2} = frac{15u - 15}{u^2})So, the equation becomes:(4 - frac{8}{u} + frac{15u - 15}{u^2} = 4)Subtract 4 from both sides:(-frac{8}{u} + frac{15u - 15}{u^2} = 0)Multiply both sides by (u^2) to eliminate denominators:(-8u + 15u - 15 = 0)Combine like terms:(7u - 15 = 0)So, (7u = 15) → (u = frac{15}{7})But (u = 1 + 4k^2 = frac{15}{7}), so:(4k^2 = frac{15}{7} - 1 = frac{8}{7})Thus, (k^2 = frac{2}{7}), so (k = pm frac{sqrt{14}}{7})Now, recall that (y_0 = -frac{6k}{1 + 4k^2}). Since (1 + 4k^2 = u = frac{15}{7}), we have:(y_0 = -frac{6k}{frac{15}{7}} = -frac{6k times 7}{15} = -frac{42k}{15} = -frac{14k}{5})Since (k = pm frac{sqrt{14}}{7}), substituting:(y_0 = -frac{14}{5} times left(pm frac{sqrt{14}}{7}right) = mp frac{14 times sqrt{14}}{35} = mp frac{2sqrt{14}}{5})So, (y_0 = pm frac{2sqrt{14}}{5})Wait, but earlier, when we considered the case when (k = 0), we might have another solution. Let me check.If (k = 0), then the line (l) is horizontal, passing through ((-2, 0)). So, the equation is (y = 0). The other intersection point (B) would be at ((2, 0)). So, points (A(-2, 0)) and (B(2, 0)). The midpoint (M) is ((0, 0)). The perpendicular bisector would be the y-axis, since the segment is horizontal. So, point (Q(0, y_0)) lies on the y-axis, which is the perpendicular bisector.Now, compute (overrightarrow{QA} cdot overrightarrow{QB}).(overrightarrow{QA} = (-2, -y_0))(overrightarrow{QB} = (2, -y_0))Dot product: ((-2)(2) + (-y_0)(-y_0) = -4 + y_0^2)Set this equal to 4:(-4 + y_0^2 = 4) → (y_0^2 = 8) → (y_0 = pm 2sqrt{2})So, in this case, (y_0 = pm 2sqrt{2})Therefore, combining both cases, (y_0) can be (pm 2sqrt{2}) or (pm frac{2sqrt{14}}{5})But wait, in part (i), we found that (k = pm 1), which led to specific values for (y_0). However, in part (ii), we considered both (k = 0) and (k neq 0). So, the possible values for (y_0) are both (pm 2sqrt{2}) and (pm frac{2sqrt{14}}{5}).But let me verify if (k = 0) is a valid case. When (k = 0), the line is horizontal, and the distance between (A) and (B) is 4, which is greater than (frac{4sqrt{2}}{5}). But in part (i), the distance is given as (frac{4sqrt{2}}{5}), so (k = 0) is not a solution for part (i). However, in part (ii), the condition is separate, so (k = 0) is a valid case for part (ii). Therefore, both sets of solutions are valid for part (ii).So, the possible values for (y_0) are (pm 2sqrt{2}) and (pm frac{2sqrt{14}}{5})**Final Answer**Ⅰ. The equation of the ellipse is (boxed{dfrac{x^2}{4} + y^2 = 1}).Ⅱ. (i) The slope angles of line (l) are (boxed{dfrac{pi}{4}}) and (boxed{dfrac{3pi}{4}}).Ⅱ. (ii) The value of (y_0) is (boxed{pm 2sqrt{2}}) or (boxed{pm dfrac{2sqrt{14}}{5}}).