💡Okay, so I have this problem here where I need to find the minimum value of ( a + 2b ) given that ( a > 0 ), ( b > 0 ), and they satisfy the equation ( 2a + b - ab = 0 ). The options are A: 9, B: ( 5 + 2sqrt{2} ), C: 8, and D: ( 7sqrt{2} ). Hmm, let me try to figure this out step by step.First, I need to understand the given equation: ( 2a + b - ab = 0 ). Maybe I can rearrange this equation to express one variable in terms of the other. Let me try that.Starting with:[ 2a + b - ab = 0 ]I can rearrange this to:[ 2a + b = ab ]Hmm, maybe I can factor this or express it differently. Let's see. If I move all terms to one side:[ ab - 2a - b = 0 ]Hmm, that doesn't seem immediately helpful. Maybe I can factor by grouping. Let me try adding 2 to both sides to see if that helps:[ ab - 2a - b + 2 = 2 ]Now, let's factor the left side:[ (a - 1)(b - 2) = 2 ]Oh, that's interesting! So, ( (a - 1)(b - 2) = 2 ). That might be a useful form. So, this tells me that ( a - 1 ) and ( b - 2 ) are positive since ( a > 0 ) and ( b > 0 ). Therefore, ( a > 1 ) and ( b > 2 ).Now, I need to find the minimum value of ( a + 2b ). Maybe I can express ( a ) in terms of ( b ) or vice versa using the equation ( (a - 1)(b - 2) = 2 ).Let me solve for ( a ) in terms of ( b ):[ (a - 1)(b - 2) = 2 ][ a - 1 = frac{2}{b - 2} ][ a = frac{2}{b - 2} + 1 ]So, ( a = frac{2}{b - 2} + 1 ). Now, let's substitute this into the expression ( a + 2b ):[ a + 2b = left( frac{2}{b - 2} + 1 right) + 2b ][ = frac{2}{b - 2} + 1 + 2b ]Hmm, this looks a bit complicated. Maybe I can simplify it further or find a way to minimize it. Let me write it as:[ a + 2b = 2b + 1 + frac{2}{b - 2} ]Now, I need to find the minimum value of this expression with respect to ( b ). Since ( b > 2 ), I can let ( t = b - 2 ), so ( t > 0 ). Then, ( b = t + 2 ), and substituting back:[ a + 2b = 2(t + 2) + 1 + frac{2}{t} ][ = 2t + 4 + 1 + frac{2}{t} ][ = 2t + 5 + frac{2}{t} ]So, now the expression is ( 2t + 5 + frac{2}{t} ). I need to find the minimum of this expression for ( t > 0 ). This seems more manageable.I recall that for expressions of the form ( kt + frac{m}{t} ), the minimum can be found using the AM-GM inequality, which states that for positive numbers, the arithmetic mean is greater than or equal to the geometric mean. Specifically, ( kt + frac{m}{t} geq 2sqrt{km} ).Let me apply this to the terms ( 2t ) and ( frac{2}{t} ):[ 2t + frac{2}{t} geq 2sqrt{2t cdot frac{2}{t}} ][ = 2sqrt{4} ][ = 2 times 2 ][ = 4 ]So, ( 2t + frac{2}{t} geq 4 ). Therefore, the entire expression ( 2t + 5 + frac{2}{t} ) is greater than or equal to ( 4 + 5 = 9 ).So, the minimum value of ( a + 2b ) is 9. Let me check if this minimum is achievable. For equality in AM-GM, we need ( 2t = frac{2}{t} ), which implies ( t^2 = 1 ), so ( t = 1 ) since ( t > 0 ).Therefore, ( t = 1 ) gives ( b = t + 2 = 3 ). Then, substituting back into ( a = frac{2}{b - 2} + 1 ):[ a = frac{2}{3 - 2} + 1 = 2 + 1 = 3 ]So, when ( a = 3 ) and ( b = 3 ), the expression ( a + 2b = 3 + 6 = 9 ), which matches our earlier result.Let me verify if this satisfies the original equation:[ 2a + b - ab = 2(3) + 3 - (3)(3) = 6 + 3 - 9 = 0 ]Yes, it does satisfy the equation.Just to be thorough, let me check if any of the other options could be lower. Option B is ( 5 + 2sqrt{2} ), which is approximately ( 5 + 2.828 = 7.828 ), which is less than 9. But since we found that the minimum is 9, this suggests that maybe I made a mistake somewhere.Wait, hold on. Let me double-check my steps. I transformed the original equation into ( (a - 1)(b - 2) = 2 ), then expressed ( a ) in terms of ( b ), substituted into ( a + 2b ), and then made a substitution ( t = b - 2 ). Then, I applied AM-GM to ( 2t + frac{2}{t} ), which gave me a minimum of 4, leading to a total of 9.Is there another way to approach this problem? Maybe using calculus? Let me try that.Let me consider ( a + 2b ) as a function to minimize, subject to the constraint ( 2a + b - ab = 0 ). I can use Lagrange multipliers or substitution. Since I already expressed ( a ) in terms of ( b ), substitution seems straightforward.We have ( a = frac{2}{b - 2} + 1 ), so ( a + 2b = frac{2}{b - 2} + 1 + 2b ). Let me denote this as ( f(b) = 2b + 1 + frac{2}{b - 2} ).To find the minimum, take the derivative of ( f(b) ) with respect to ( b ) and set it to zero.First, compute ( f'(b) ):[ f'(b) = 2 + 0 + frac{d}{db}left( frac{2}{b - 2} right) ][ = 2 + frac{-2}{(b - 2)^2} ]Set ( f'(b) = 0 ):[ 2 - frac{2}{(b - 2)^2} = 0 ][ 2 = frac{2}{(b - 2)^2} ][ (b - 2)^2 = 1 ][ b - 2 = pm 1 ]Since ( b > 2 ), we take the positive root:[ b - 2 = 1 ][ b = 3 ]So, ( b = 3 ), and substituting back into ( a = frac{2}{b - 2} + 1 ):[ a = frac{2}{1} + 1 = 3 ]Thus, ( a = 3 ) and ( b = 3 ), giving ( a + 2b = 9 ). So, calculus confirms the earlier result.Wait, but option B is ( 5 + 2sqrt{2} ), which is approximately 7.828, which is less than 9. Maybe I made a mistake in the substitution or the AM-GM step.Let me revisit the substitution. I had ( a + 2b = 2t + 5 + frac{2}{t} ), where ( t = b - 2 ). Then, I applied AM-GM to ( 2t + frac{2}{t} ), getting a minimum of 4, so total is 9.But perhaps I should consider the entire expression ( 2t + frac{2}{t} ) and see if there's a better way to apply AM-GM or another inequality.Alternatively, maybe I can write ( 2t + frac{2}{t} = 2left( t + frac{1}{t} right) ). Then, since ( t + frac{1}{t} geq 2 ) by AM-GM, multiplying by 2 gives ( 2left( t + frac{1}{t} right) geq 4 ), which is consistent with what I had before.So, that still gives a minimum of 4, leading to 9. So, I think my initial approach is correct.Wait, but let me check if ( a ) and ( b ) can take other values that might give a lower sum. For example, suppose ( a = 2 ), then from the original equation ( 2(2) + b - 2b = 0 implies 4 + b - 2b = 0 implies 4 - b = 0 implies b = 4 ). Then, ( a + 2b = 2 + 8 = 10 ), which is higher than 9.If ( a = 4 ), then ( 2(4) + b - 4b = 0 implies 8 + b - 4b = 0 implies 8 - 3b = 0 implies b = 8/3 approx 2.666 ). Then, ( a + 2b = 4 + 16/3 approx 4 + 5.333 = 9.333 ), which is still higher than 9.If ( a = 1 ), but ( a > 1 ) as per earlier, so ( a = 1 ) is not allowed. If ( a = 1.5 ), then ( 2(1.5) + b - 1.5b = 0 implies 3 + b - 1.5b = 0 implies 3 - 0.5b = 0 implies b = 6 ). Then, ( a + 2b = 1.5 + 12 = 13.5 ), which is way higher.Alternatively, if ( b = 4 ), then from ( (a - 1)(4 - 2) = 2 implies (a - 1)(2) = 2 implies a - 1 = 1 implies a = 2 ). So, ( a + 2b = 2 + 8 = 10 ), again higher.Wait, maybe trying specific values isn't helping. Let me think differently. Perhaps I can use substitution with ( a = frac{2}{b - 2} + 1 ) and then express ( a + 2b ) as a function of ( b ), then find its minimum.We have:[ a + 2b = frac{2}{b - 2} + 1 + 2b ]Let me denote ( f(b) = 2b + 1 + frac{2}{b - 2} )To find the minimum, take the derivative:[ f'(b) = 2 - frac{2}{(b - 2)^2} ]Set ( f'(b) = 0 ):[ 2 - frac{2}{(b - 2)^2} = 0 ][ 2 = frac{2}{(b - 2)^2} ][ (b - 2)^2 = 1 ][ b - 2 = 1 ] (since ( b > 2 ))[ b = 3 ]So, ( b = 3 ), then ( a = frac{2}{3 - 2} + 1 = 3 ). Thus, ( a + 2b = 3 + 6 = 9 ).This confirms again that the minimum is 9. So, why is option B ( 5 + 2sqrt{2} ) an option? Maybe I need to check if there's another approach where this could be the minimum.Alternatively, perhaps I can use the method of Lagrange multipliers.Let me set up the function to minimize: ( f(a, b) = a + 2b )Subject to the constraint: ( g(a, b) = 2a + b - ab = 0 )The method of Lagrange multipliers tells us that at the minimum, the gradient of ( f ) is proportional to the gradient of ( g ). So:[ nabla f = lambda nabla g ]Compute the gradients:[ nabla f = (1, 2) ][ nabla g = (2 - b, 1 - a) ]So, setting up the equations:[ 1 = lambda (2 - b) ][ 2 = lambda (1 - a) ]From the first equation:[ lambda = frac{1}{2 - b} ]From the second equation:[ lambda = frac{2}{1 - a} ]Set them equal:[ frac{1}{2 - b} = frac{2}{1 - a} ]Cross-multiplying:[ 1 - a = 2(2 - b) ][ 1 - a = 4 - 2b ][ -a + 2b = 3 ][ 2b - a = 3 ]Now, we have two equations:1. ( 2a + b - ab = 0 )2. ( 2b - a = 3 )Let me solve equation 2 for ( a ):[ a = 2b - 3 ]Substitute into equation 1:[ 2(2b - 3) + b - (2b - 3)b = 0 ][ 4b - 6 + b - (2b^2 - 3b) = 0 ][ 5b - 6 - 2b^2 + 3b = 0 ][ -2b^2 + 8b - 6 = 0 ]Multiply both sides by -1:[ 2b^2 - 8b + 6 = 0 ]Divide by 2:[ b^2 - 4b + 3 = 0 ]Factor:[ (b - 1)(b - 3) = 0 ]So, ( b = 1 ) or ( b = 3 )But ( b > 2 ), so ( b = 3 ). Then, ( a = 2(3) - 3 = 3 ). So, again, ( a = 3 ), ( b = 3 ), and ( a + 2b = 9 ).This method also leads to the same conclusion. So, I think 9 is indeed the minimum value.Wait, but let me check if there's a way to get a lower value. Maybe if I consider the expression ( a + 2b ) and use Cauchy-Schwarz or another inequality.Alternatively, perhaps I can express ( a + 2b ) in terms of ( (a - 1) ) and ( (b - 2) ) since we have ( (a - 1)(b - 2) = 2 ).Let me denote ( x = a - 1 ) and ( y = b - 2 ), so ( xy = 2 ). Then, ( a = x + 1 ) and ( b = y + 2 ). So, ( a + 2b = (x + 1) + 2(y + 2) = x + 1 + 2y + 4 = x + 2y + 5 ).So, we need to minimize ( x + 2y + 5 ) subject to ( xy = 2 ), where ( x > 0 ) and ( y > 0 ).This seems similar to the earlier problem. Let me express ( x ) in terms of ( y ): ( x = frac{2}{y} ). Then, the expression becomes:[ frac{2}{y} + 2y + 5 ]Now, let me denote this as ( f(y) = frac{2}{y} + 2y + 5 ). To find the minimum, take the derivative with respect to ( y ):[ f'(y) = -frac{2}{y^2} + 2 ]Set ( f'(y) = 0 ):[ -frac{2}{y^2} + 2 = 0 ][ 2 = frac{2}{y^2} ][ y^2 = 1 ][ y = 1 ] (since ( y > 0 ))So, ( y = 1 ), then ( x = frac{2}{1} = 2 ). Therefore, ( a = x + 1 = 3 ), ( b = y + 2 = 3 ), and ( a + 2b = 9 ).Again, the same result. It seems consistent across different methods: substitution, calculus, Lagrange multipliers, and even variable substitution with inequalities.So, I think the minimum value is indeed 9, which is option A.Just to be absolutely sure, let me check if there's any possibility of a lower value by considering the expression ( a + 2b ) and the constraint ( 2a + b = ab ).From the constraint, ( ab - 2a - b = 0 ). Let me add 2 to both sides:[ ab - 2a - b + 2 = 2 ]Which factors to:[ (a - 1)(b - 2) = 2 ]As before, so ( a = 1 + frac{2}{b - 2} ). Then, ( a + 2b = 1 + frac{2}{b - 2} + 2b ). Let me consider this as a function of ( b ), say ( f(b) = 2b + 1 + frac{2}{b - 2} ).Taking the derivative:[ f'(b) = 2 - frac{2}{(b - 2)^2} ]Setting to zero:[ 2 = frac{2}{(b - 2)^2} ][ (b - 2)^2 = 1 ][ b - 2 = 1 ][ b = 3 ]Then, ( a = 1 + frac{2}{1} = 3 ), so ( a + 2b = 9 ).All methods converge to the same answer. Therefore, I'm confident that the minimum value is 9, which is option A.**Final Answer**boxed{A}

💡First, I recognize that the outdoor section of the restaurant is rectangular, so I can use the formula for the area of a rectangle, which is Area = Width × Length.I know the area of the outdoor section is 24 square feet and the width is 4 feet. Plugging these values into the formula gives me:24 = 4 × LengthTo find the length, I need to isolate it in the equation. I can do this by dividing both sides of the equation by the width, which is 4 feet:Length = 24 / 4Calculating this gives me:Length = 6 feetTherefore, the length of the outdoor section is 6 feet.

💡Alright, so I have this geometry problem here about a parallelogram ABCD. Let me try to visualize it first. In a parallelogram, opposite sides are equal and parallel, right? So AB is equal to CD, and AD is equal to BC. They've given that AB is 1 unit long. Now, there's a point K on side AD such that KD is 1 unit. So, if I imagine AD, K is somewhere on it, and the segment from K to D is 1 unit. That means AK must be AD minus 1, right? So if I let AD be some length, say, x, then AK would be x - 1. Hmm, but wait, I don't know AD yet, that's what I need to find. Maybe I can denote AD as something else, like y, so that AK is y - 1. They also mention two angles: angle ABK is 90 degrees, and angle DBK is 30 degrees. Okay, so point K is connected to B, forming triangle ABK and triangle DBK. Since angle ABK is 90 degrees, triangle ABK is a right-angled triangle. And angle DBK is 30 degrees, which is a nice angle because it relates to 30-60-90 triangles, which have known side ratios.Let me try to draw this out mentally. Parallelogram ABCD, AB is 1, so AB is the base. AD is the side we need to find. Point K is on AD, closer to D since KD is 1. So from K, we draw lines to B, creating two angles at B: one is 90 degrees with AB, and the other is 30 degrees with DB.Since ABCD is a parallelogram, the diagonals bisect each other. So the diagonals AC and BD intersect at the midpoint. But I'm not sure if that's directly useful here. Maybe I can use some trigonometry or coordinate geometry.Let me try coordinate geometry. Let's place point A at the origin (0,0). Since AB is 1, point B would be at (1,0). Since it's a parallelogram, point D would be at (0, h), where h is the height, which is related to AD. Wait, but AD is the side, not necessarily the height. Hmm, maybe I need to adjust that.Alternatively, let's assign coordinates more carefully. Let me set point A at (0,0). Since AB is 1, point B is at (1,0). Now, since ABCD is a parallelogram, point D must be at (a,b), and point C would then be at (1+a, b). Now, point K is on AD, which goes from A(0,0) to D(a,b). Since KD is 1, K is 1 unit away from D. So the coordinates of K can be found by moving 1 unit from D towards A.To find K, I can parameterize AD. The vector from A to D is (a,b), so the parametric equation of AD is (ta, tb) for t from 0 to 1. The length of AD is sqrt(a² + b²). Since KD is 1, the distance from K to D is 1. So, the distance from K to D is sqrt((a - ta)² + (b - tb)²) = sqrt((a(1 - t))² + (b(1 - t))²) = (1 - t)sqrt(a² + b²) = 1. Therefore, (1 - t)sqrt(a² + b²) = 1, so t = 1 - 1/sqrt(a² + b²). Therefore, the coordinates of K are (ta, tb) = (a(1 - 1/sqrt(a² + b²)), b(1 - 1/sqrt(a² + b²))). Hmm, that seems a bit complicated, but maybe manageable.Now, angle ABK is 90 degrees. So, the lines AB and BK are perpendicular. The slope of AB is (0 - 0)/(1 - 0) = 0, so it's horizontal. Therefore, BK must be vertical if they are perpendicular. Wait, but BK is from B(1,0) to K(a(1 - 1/sqrt(a² + b²)), b(1 - 1/sqrt(a² + b²))). For BK to be vertical, the x-coordinate of K must be equal to the x-coordinate of B, which is 1. So, a(1 - 1/sqrt(a² + b²)) = 1. Let me write that down: a(1 - 1/sqrt(a² + b²)) = 1. Let me denote sqrt(a² + b²) as L, which is the length of AD. So, a(1 - 1/L) = 1. Therefore, a = 1 / (1 - 1/L) = L / (L - 1). So, a = L / (L - 1). Also, since ABCD is a parallelogram, the coordinates of C are (1 + a, b). Now, the diagonal BD goes from B(1,0) to D(a,b). The angle DBK is 30 degrees. So, the angle between BK and BD is 30 degrees. Let me find the vectors for BK and BD. Vector BK is K - B = (a(1 - 1/L) - 1, b(1 - 1/L) - 0) = (a(1 - 1/L) - 1, b(1 - 1/L)). Vector BD is D - B = (a - 1, b - 0) = (a - 1, b). The angle between BK and BD is 30 degrees, so the dot product formula can be used. The dot product of BK and BD is equal to |BK||BD|cos(30°). First, let's compute the dot product:BK • BD = [a(1 - 1/L) - 1](a - 1) + [b(1 - 1/L)]bLet me compute each term:First term: [a(1 - 1/L) - 1](a - 1) = [a - a/L - 1](a - 1)= (a - 1 - a/L)(a - 1)= (a - 1)(a - 1) - (a/L)(a - 1)= (a - 1)^2 - (a(a - 1))/LSecond term: [b(1 - 1/L)]b = b²(1 - 1/L)So, total dot product:= (a - 1)^2 - (a(a - 1))/L + b²(1 - 1/L)Now, let's compute |BK| and |BD|.|BK| is the distance from B to K:= sqrt([a(1 - 1/L) - 1]^2 + [b(1 - 1/L)]^2)Let me factor out (1 - 1/L):= sqrt([(a(1 - 1/L) - 1)^2 + (b(1 - 1/L))^2])But from earlier, we have a(1 - 1/L) = 1, so:= sqrt([1 - 1]^2 + [b(1 - 1/L)]^2) = sqrt(0 + [b(1 - 1/L)]^2) = |b(1 - 1/L)|Since lengths are positive, |BK| = b(1 - 1/L)Wait, that seems interesting. So, |BK| = b(1 - 1/L)Now, |BD| is the distance from B to D:= sqrt((a - 1)^2 + b^2)So, putting it all together, the dot product is equal to |BK||BD|cos(30°):(a - 1)^2 - (a(a - 1))/L + b²(1 - 1/L) = [b(1 - 1/L)] * sqrt((a - 1)^2 + b^2) * (√3 / 2)Hmm, this is getting quite involved. Maybe I can find expressions for a and b in terms of L.Earlier, we had a = L / (L - 1). Let me write that down: a = L / (L - 1). Also, since AD is length L, which is sqrt(a² + b²), so:L = sqrt(a² + b²)So, L² = a² + b²We can express b² = L² - a²Since a = L / (L - 1), then a² = L² / (L - 1)^2So, b² = L² - L² / (L - 1)^2 = L² [1 - 1 / (L - 1)^2] = L² [( (L - 1)^2 - 1 ) / (L - 1)^2] = L² [ (L² - 2L + 1 - 1 ) / (L - 1)^2 ] = L² [ (L² - 2L ) / (L - 1)^2 ] = L² * L(L - 2) / (L - 1)^2 = L³(L - 2) / (L - 1)^2Wait, that seems a bit messy. Maybe I can plug a = L / (L - 1) into the dot product equation.Let me recall that the dot product was:(a - 1)^2 - (a(a - 1))/L + b²(1 - 1/L) = [b(1 - 1/L)] * sqrt((a - 1)^2 + b^2) * (√3 / 2)Let me substitute a = L / (L - 1) into (a - 1):a - 1 = (L / (L - 1)) - 1 = (L - (L - 1)) / (L - 1) = 1 / (L - 1)So, (a - 1)^2 = 1 / (L - 1)^2Similarly, a(a - 1) = (L / (L - 1)) * (1 / (L - 1)) = L / (L - 1)^2So, the first two terms of the dot product:(a - 1)^2 - (a(a - 1))/L = [1 / (L - 1)^2] - [L / (L - 1)^2] / L = [1 / (L - 1)^2] - [1 / (L - 1)^2] = 0Wait, that's interesting. So, the first two terms cancel out.Now, the third term is b²(1 - 1/L). So, the entire dot product is b²(1 - 1/L)So, the equation becomes:b²(1 - 1/L) = [b(1 - 1/L)] * sqrt((a - 1)^2 + b^2) * (√3 / 2)Let me simplify both sides. First, divide both sides by b(1 - 1/L), assuming b ≠ 0 and 1 - 1/L ≠ 0, which they aren't because L > 1 (since a = L / (L - 1) must be positive and greater than 1 if L > 1).So, dividing both sides:b = sqrt((a - 1)^2 + b^2) * (√3 / 2)Let me square both sides to eliminate the square root:b² = [ (a - 1)^2 + b² ] * (3/4)Multiply both sides by 4:4b² = 3(a - 1)^2 + 3b²Subtract 3b² from both sides:b² = 3(a - 1)^2But from earlier, we have b² = L² - a²And (a - 1)^2 = 1 / (L - 1)^2So, substituting:L² - a² = 3 * [1 / (L - 1)^2]But a = L / (L - 1), so a² = L² / (L - 1)^2Thus:L² - L² / (L - 1)^2 = 3 / (L - 1)^2Multiply both sides by (L - 1)^2 to eliminate denominators:L²(L - 1)^2 - L² = 3Factor L²:L²[ (L - 1)^2 - 1 ] = 3Compute (L - 1)^2 - 1:= L² - 2L + 1 - 1 = L² - 2LSo:L²(L² - 2L) = 3Expand:L⁴ - 2L³ - 3 = 0So, we have a quartic equation: L⁴ - 2L³ - 3 = 0Hmm, solving quartic equations can be tricky. Maybe I can factor it or find rational roots.Let me try rational root theorem. Possible rational roots are ±1, ±3.Testing L=1: 1 - 2 - 3 = -4 ≠ 0L=3: 81 - 54 - 3 = 24 ≠ 0L=-1: 1 + 2 - 3 = 0 → Wait, L=-1 is a root.So, (L + 1) is a factor. Let's perform polynomial division or factor it out.Divide L⁴ - 2L³ - 3 by (L + 1):Using synthetic division:-1 | 1 -2 0 0 -3 -1 3 -3 3 1 -3 3 -3 0So, the quartic factors as (L + 1)(L³ - 3L² + 3L - 3) = 0So, the real roots are L = -1 and roots of L³ - 3L² + 3L - 3 = 0Since length can't be negative, L = -1 is discarded.Now, solve L³ - 3L² + 3L - 3 = 0Let me try rational roots again: possible roots are ±1, ±3.Testing L=1: 1 - 3 + 3 - 3 = -2 ≠ 0L=3: 27 - 27 + 9 - 3 = 6 ≠ 0L=√3: Let me see, approximate value. Maybe it's a real root.Let me compute f(2): 8 - 12 + 6 - 3 = -1f(3): 27 - 27 + 9 - 3 = 6So, there's a root between 2 and 3.Using Newton-Raphson method:f(L) = L³ - 3L² + 3L - 3f'(L) = 3L² - 6L + 3Take L₀ = 2f(2) = 8 - 12 + 6 - 3 = -1f'(2) = 12 - 12 + 3 = 3Next approximation: L₁ = 2 - f(2)/f'(2) = 2 - (-1)/3 = 2 + 1/3 ≈ 2.333Compute f(2.333):≈ (2.333)^3 - 3*(2.333)^2 + 3*(2.333) - 3≈ 12.703 - 16.666 + 7 - 3 ≈ 0.037f'(2.333) ≈ 3*(2.333)^2 - 6*(2.333) + 3 ≈ 16.666 - 14 + 3 ≈ 5.666Next approximation: L₂ = 2.333 - 0.037 / 5.666 ≈ 2.333 - 0.0065 ≈ 2.3265Compute f(2.3265):≈ (2.3265)^3 - 3*(2.3265)^2 + 3*(2.3265) - 3≈ 12.55 - 16.12 + 7 - 3 ≈ 0.43Wait, that seems inconsistent. Maybe my calculations are off. Alternatively, perhaps the exact root is the cube root of something.Wait, the equation is L³ - 3L² + 3L - 3 = 0Let me make a substitution: Let M = L - 1Then, L = M + 1Substitute into the equation:(M + 1)^3 - 3(M + 1)^2 + 3(M + 1) - 3 = 0Expand:M³ + 3M² + 3M + 1 - 3(M² + 2M + 1) + 3M + 3 - 3 = 0Simplify term by term:M³ + 3M² + 3M + 1 - 3M² - 6M - 3 + 3M + 3 - 3 = 0Combine like terms:M³ + (3M² - 3M²) + (3M - 6M + 3M) + (1 - 3 + 3 - 3) = 0Simplify:M³ + 0M² + 0M + (-2) = 0So, M³ - 2 = 0 → M³ = 2 → M = ∛2Therefore, L = M + 1 = ∛2 + 1So, the length AD is L = 1 + ∛2That seems to be the solution. Let me check if this makes sense.If L = 1 + ∛2, then a = L / (L - 1) = (1 + ∛2) / ∛2 = (1/∛2) + 1And b² = L² - a²Compute L² = (1 + ∛2)^2 = 1 + 2∛2 + (∛2)^2a² = ( (1 + ∛2)/∛2 )² = (1 + 2∛2 + (∛2)^2 ) / (∛2)^2 = (1 + 2∛2 + 2^(2/3)) / 2^(2/3)Hmm, this is getting complicated, but since we arrived at L = 1 + ∛2 through substitution and solving the quartic, which reduced to a cubic that we could solve by substitution, I think this is correct.So, the length AD is 1 + ∛2.

💡First, I need to determine the total number of light bulbs Reyna has. Since there are 20 lamps and each lamp contains 7 light bulbs, I can calculate the total by multiplying the number of lamps by the number of light bulbs per lamp.Next, I should find out how many lamps have burnt-out light bulbs. The problem states that 1/4 of the lamps have 2 burnt-out light bulbs each. To find this, I'll calculate 1/4 of 20 lamps and then multiply that number by 2 to get the total number of burnt-out light bulbs.Finally, to find the number of working light bulbs, I'll subtract the total number of burnt-out light bulbs from the total number of light bulbs.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that {a_n} is an arithmetic sequence, and S_n is the sum of the first n terms. They give me that a_1, which is the first term, is 1, and S_3, the sum of the first three terms, is 9. I need to find the common difference d of this sequence. The options are A:1, B:2, C:3, D:4.Alright, let's start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d. So, if I know the first term and the common difference, I can find any term in the sequence.The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d. So, for example, the second term a_2 would be a_1 + d, the third term a_3 would be a_1 + 2d, and so on.Now, the problem also mentions S_n, which is the sum of the first n terms. I remember that the formula for the sum of the first n terms of an arithmetic sequence is S_n = n/2 * (2a_1 + (n-1)d). Alternatively, it can also be written as S_n = n*(a_1 + a_n)/2 because the average of the first and nth term multiplied by the number of terms gives the sum.Given that S_3 = 9, let's plug in the values we know into the sum formula. So, n is 3, a_1 is 1, and S_3 is 9. Let me use the first formula: S_n = n/2 * (2a_1 + (n-1)d). Plugging in the values, we get:9 = 3/2 * (2*1 + (3-1)d)Let me simplify this step by step. First, calculate 2*1, which is 2. Then, (3-1)d is 2d. So inside the parentheses, it's 2 + 2d. Now, multiply that by 3/2:9 = (3/2)*(2 + 2d)Let me compute (3/2)*(2 + 2d). Multiplying 3/2 by 2 gives 3, and 3/2 multiplied by 2d gives 3d. So, the equation becomes:9 = 3 + 3dNow, I can solve for d. Subtract 3 from both sides:9 - 3 = 3d6 = 3dDivide both sides by 3:d = 2So, the common difference d is 2. Let me check if this makes sense. If a_1 is 1, then a_2 would be 1 + 2 = 3, and a_3 would be 1 + 2*2 = 5. The sum of the first three terms is 1 + 3 + 5 = 9, which matches S_3 = 9. So, that checks out.Alternatively, I could have used the second formula for the sum, S_n = n*(a_1 + a_n)/2. Let's try that as a verification. We know S_3 = 9, n = 3, a_1 = 1, so we need to find a_3.From the nth term formula, a_3 = a_1 + 2d = 1 + 2d. Then, S_3 = 3*(1 + a_3)/2 = 9. Plugging in a_3:3*(1 + 1 + 2d)/2 = 9Simplify inside the parentheses: 1 + 1 + 2d = 2 + 2dSo, 3*(2 + 2d)/2 = 9Multiply 3 by (2 + 2d)/2: (3*(2 + 2d))/2 = 9Which simplifies to (6 + 6d)/2 = 9Divide numerator terms by 2: 3 + 3d = 9Again, subtract 3 from both sides: 3d = 6Divide by 3: d = 2Same result. That gives me more confidence that d = 2 is correct.Let me also think about what each term would be with d = 2. The sequence would be 1, 3, 5, 7, 9, ... Each time adding 2. The sum of the first three terms is 1 + 3 + 5 = 9, which is exactly what the problem states. So, everything lines up.I don't think I made any mistakes here. The calculations are straightforward, and both methods give the same answer. So, I'm pretty sure the common difference d is 2, which is option B.**Final Answer**The common difference ( d ) is boxed{B}.

💡Okay, so I have this problem here: If x is a perfect square, what is the expression for the perfect square after the next larger perfect square? The options are A) x + 2√x + 1, B) x + 4√x + 1, C) x + 4√x + 4, D) x + 8√x + 16, and E) x + 2√x + 4.Hmm, let me try to understand what it's asking. It says x is a perfect square, so that means x is equal to some integer squared. Let's say x = k², where k is an integer. Now, they want the expression for the perfect square that comes after the next larger one. So, if x is k², the next larger perfect square would be (k+1)², and the one after that would be (k+2)². So, they want an expression for (k+2)² in terms of x.Alright, so let's write that out. If x = k², then (k+2)² is equal to k² + 4k + 4. But since x is k², I can substitute that in. So, (k+2)² = x + 4k + 4. Now, I need to express this in terms of x without k. Since x = k², then k is equal to the square root of x, right? So, k = √x. Therefore, 4k is 4√x. So, substituting that back in, (k+2)² = x + 4√x + 4.Looking at the options, that's option C: x + 4√x + 4. Let me double-check to make sure I didn't make a mistake. If x is a perfect square, say x = 9, which is 3². Then the next larger perfect square would be 16, and the one after that would be 25. Let's see if the expression works. Plugging x = 9 into option C: 9 + 4√9 + 4 = 9 + 4*3 + 4 = 9 + 12 + 4 = 25. That's correct. Let me try another one. If x = 16, which is 4², then the next larger perfect square is 25, and the one after that is 36. Plugging into option C: 16 + 4√16 + 4 = 16 + 4*4 + 4 = 16 + 16 + 4 = 36. Perfect, that works too.Just to be thorough, let me check another option to make sure it's not a coincidence. Let's try x = 25. The next larger perfect square is 36, and the one after that is 49. Using option C: 25 + 4√25 + 4 = 25 + 4*5 + 4 = 25 + 20 + 4 = 49. Yep, that's correct as well. So, it seems like option C is the right answer.I think I've got it. The key was to recognize that x is a perfect square, so it can be expressed as k², and then the next next perfect square would be (k+2)². Expanding that and substituting back in terms of x gives the expression in option C.