💡First, I need to identify the largest and the second largest numbers from the given lottery winning numbers: 11, 25, 42, 15, 28, and 39.To do this, I will arrange the numbers in ascending order:11, 15, 25, 28, 39, 42.From this ordered list, the largest number is 42, and the second largest number is 39.Next, I will calculate the difference between the largest number and the second largest number:42 - 39 = 3.Therefore, the difference between the largest and the second largest number is 3.

💡Okay, so I have this problem here: If the exponential function ( y = (2a - 1)^x ) is strictly decreasing on ( mathbb{R} ), then what is the range of values for ( a )? The options are A: ( (0,1) ), B: ( (frac{1}{2}, +infty) ), C: ( (frac{1}{2}, 1) ), and D: ( (1, +infty) ).Hmm, I remember that exponential functions can be either increasing or decreasing depending on their base. If the base is greater than 1, the function is increasing, and if the base is between 0 and 1, the function is decreasing. So, since this function is strictly decreasing, the base must be between 0 and 1.The base here is ( (2a - 1) ). So, I need to set up an inequality where ( 0 < (2a - 1) < 1 ). Let me write that down:( 0 < 2a - 1 < 1 )Now, I need to solve this compound inequality for ( a ). Let me break it into two parts.First, the left part: ( 0 < 2a - 1 ). If I add 1 to both sides, I get:( 1 < 2a )Then, dividing both sides by 2:( frac{1}{2} < a )Okay, so ( a ) has to be greater than ( frac{1}{2} ).Now, the right part of the inequality: ( 2a - 1 < 1 ). Adding 1 to both sides:( 2a < 2 )Dividing both sides by 2:( a < 1 )So, putting both parts together, ( a ) must satisfy:( frac{1}{2} < a < 1 )Looking back at the options, that's option C: ( (frac{1}{2}, 1) ).Wait, let me double-check to make sure I didn't make a mistake. If ( a ) is between ( frac{1}{2} ) and 1, then ( 2a - 1 ) would be between 0 and 1, right? Because if ( a = frac{1}{2} ), then ( 2a - 1 = 0 ), and if ( a = 1 ), then ( 2a - 1 = 1 ). So, as ( a ) increases from ( frac{1}{2} ) to 1, ( 2a - 1 ) increases from 0 to 1.Since the base ( (2a - 1) ) is between 0 and 1, the function ( y = (2a - 1)^x ) is indeed strictly decreasing. That makes sense because when the base is between 0 and 1, the function decreases as ( x ) increases.Let me also consider the other options to ensure none of them could work. Option A is ( (0,1) ). If ( a ) were in this range, ( 2a - 1 ) would be between ( -1 ) and 1. But since the base must be positive (as exponential functions with negative bases aren't defined for all real numbers), ( 2a - 1 ) has to be greater than 0. So, ( a ) must be greater than ( frac{1}{2} ), which is why option A isn't correct.Option B is ( (frac{1}{2}, +infty) ). If ( a ) were greater than 1, then ( 2a - 1 ) would be greater than 1, making the function ( y = (2a - 1)^x ) strictly increasing, not decreasing. So, that's not right either.Option D is ( (1, +infty) ), which, as I just thought, would result in a base greater than 1, leading to an increasing function. So, that's out.Therefore, the only correct option is C: ( (frac{1}{2}, 1) ). I think I've covered all the bases here, so I feel confident with this answer.**Final Answer**boxed{C}

💡First, I'll define the variables for the ages of Mary, Sally, and Danielle as ( m ), ( s ), and ( d ) respectively.Mary is 30% older than Sally, which means:[m = 1.3s]Sally is 30% younger than Danielle, so:[s = 0.7d]Substituting Sally's age into Mary's equation:[m = 1.3 times 0.7d = 0.91d]The sum of their ages is 25.2 years:[m + s + d = 0.91d + 0.7d + d = 2.61d = 25.2]Solving for Danielle's age:[d = frac{25.2}{2.61} approx 9.66]Calculating Mary's age:[m = 0.91 times 9.66 approx 8.78]Since Mary is approximately 8.78 years old, she will be 9 years old on her next birthday.

💡Okay, so I have this problem where I need to find the minimum value of the quadratic expression:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]over all real numbers (x) and (y). Hmm, quadratic expressions can sometimes be tricky, but I remember that completing the square is a useful method for finding minima or maxima. Let me try to apply that here.First, I should probably rearrange and group the terms in a way that makes completing the square easier. Let me see, the expression has (x^2), (xy), and (y^2) terms, as well as linear terms in (x) and (y). Maybe I can group the (x) terms together and the (y) terms together.So, let me rewrite the expression:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]I can group the (x) terms and the (y) terms:[(3x^2 + 4xy - 6x) + (2y^2 - 8y) + 6]Hmm, okay. Now, I need to complete the square for both (x) and (y). But since there's an (xy) term, it's a bit more complicated. I think I might need to treat this as a quadratic form and maybe use some method involving matrices or eigenvalues, but that might be overcomplicating things. Let me stick to completing the square for now.Looking at the (x) terms: (3x^2 + 4xy - 6x). Maybe I can factor out the coefficient of (x^2), which is 3:[3left(x^2 + frac{4}{3}xy - 2xright)]Similarly, for the (y) terms: (2y^2 - 8y). Factor out the 2:[2left(y^2 - 4yright)]So now, the expression becomes:[3left(x^2 + frac{4}{3}xy - 2xright) + 2left(y^2 - 4yright) + 6]Now, let's try to complete the square for the (x) terms inside the first parentheses. The expression is (x^2 + frac{4}{3}xy - 2x). Hmm, this still has an (xy) term, which complicates things. Maybe I should consider completing the square in terms of (x) treating (y) as a constant.Let me rewrite the (x) terms:(x^2 + left(frac{4}{3}y - 2right)x)To complete the square for (x), I need to find a term that makes this a perfect square trinomial. The general form is ((x + a)^2 = x^2 + 2ax + a^2). Comparing this to my expression, I have:(x^2 + left(frac{4}{3}y - 2right)x + text{something})So, the coefficient of (x) is (frac{4}{3}y - 2), which should be equal to (2a). Therefore, (a = frac{2}{3}y - 1). So, the square would be:[left(x + frac{2}{3}y - 1right)^2 = x^2 + left(frac{4}{3}y - 2right)x + left(frac{2}{3}y - 1right)^2]But in my expression, I only have up to the (x) term, so I need to subtract the square term to keep the expression equivalent. So, rewriting the (x) terms:[x^2 + left(frac{4}{3}y - 2right)x = left(x + frac{2}{3}y - 1right)^2 - left(frac{2}{3}y - 1right)^2]Therefore, substituting back into the expression:[3left[left(x + frac{2}{3}y - 1right)^2 - left(frac{2}{3}y - 1right)^2right] + 2left(y^2 - 4yright) + 6]Expanding this, we get:[3left(x + frac{2}{3}y - 1right)^2 - 3left(frac{2}{3}y - 1right)^2 + 2left(y^2 - 4yright) + 6]Now, let's simplify the terms that don't involve (x):First, expand (-3left(frac{2}{3}y - 1right)^2):[-3left(frac{4}{9}y^2 - frac{4}{3}y + 1right) = -frac{4}{3}y^2 + 4y - 3]Next, expand (2left(y^2 - 4yright)):[2y^2 - 8y]Now, combine all these terms:[-frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Combine like terms:For (y^2): (-frac{4}{3}y^2 + 2y^2 = left(-frac{4}{3} + frac{6}{3}right)y^2 = frac{2}{3}y^2)For (y): (4y - 8y = -4y)Constants: (-3 + 6 = 3)So, the expression simplifies to:[frac{2}{3}y^2 - 4y + 3]Now, let's write the entire expression:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}y^2 - 4y + 3]Now, I need to complete the square for the remaining (y) terms: (frac{2}{3}y^2 - 4y + 3). Let's factor out (frac{2}{3}):[frac{2}{3}left(y^2 - 6yright) + 3]Now, complete the square inside the parentheses:(y^2 - 6y)The square completion would be:[(y - 3)^2 = y^2 - 6y + 9]So, we have:(y^2 - 6y = (y - 3)^2 - 9)Substituting back:[frac{2}{3}left[(y - 3)^2 - 9right] + 3 = frac{2}{3}(y - 3)^2 - 6 + 3 = frac{2}{3}(y - 3)^2 - 3]So, now, the entire expression becomes:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3]Therefore, the original expression is:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3 + 6]Wait, no, I think I missed something. Earlier, after simplifying the terms, I had:[frac{2}{3}y^2 - 4y + 3]Which became:[frac{2}{3}(y - 3)^2 - 3]So, the entire expression is:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3 + 6]Wait, no, the original expression after substituting back was:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}y^2 - 4y + 3]Which then became:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3]But then we had the constant term from earlier, which was +6. Wait, no, let me check.Wait, no, the original expression after expanding was:[3left(x + frac{2}{3}y - 1right)^2 - frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Which simplified to:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}y^2 - 4y + 3]Then, completing the square for (y):[frac{2}{3}(y - 3)^2 - 3]So, the entire expression is:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3]But wait, the original expression had a +6 at the end. So, where did that go? Let me check.Wait, no, the +6 was part of the original expression, which was included in the simplification. Let me retrace:Original expression:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]After grouping and completing the square for (x):[3left(x + frac{2}{3}y - 1right)^2 - 3left(frac{2}{3}y - 1right)^2 + 2left(y^2 - 4yright) + 6]Then, expanding the squares:[-3left(frac{4}{9}y^2 - frac{4}{3}y + 1right) = -frac{4}{3}y^2 + 4y - 3]And (2left(y^2 - 4yright) = 2y^2 - 8y)So, combining these:[-frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Which simplifies to:[frac{2}{3}y^2 - 4y + 3]Then, completing the square for (y):[frac{2}{3}(y - 3)^2 - 3]So, the entire expression is:[3left(x + frac{2}{3}y - 1right)^2 + frac{2}{3}(y - 3)^2 - 3]But wait, the original expression had a +6, and after simplifying, we ended up with -3. So, the total constants are -3, but the original had +6. Hmm, that suggests I might have made a mistake in the simplification.Wait, let's go back step by step.Original expression:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]Grouped as:[3x^2 + 4xy - 6x + 2y^2 - 8y + 6]Factored as:[3(x^2 + frac{4}{3}xy - 2x) + 2(y^2 - 4y) + 6]Completed the square for (x):[3left[(x + frac{2}{3}y - 1)^2 - (frac{2}{3}y - 1)^2right] + 2(y^2 - 4y) + 6]Expanded:[3(x + frac{2}{3}y - 1)^2 - 3(frac{4}{9}y^2 - frac{4}{3}y + 1) + 2y^2 - 8y + 6]Simplify the terms:[-3(frac{4}{9}y^2 - frac{4}{3}y + 1) = -frac{4}{3}y^2 + 4y - 3]So, combining all terms:[3(x + frac{2}{3}y - 1)^2 - frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Now, combine like terms:For (y^2): (-frac{4}{3}y^2 + 2y^2 = frac{2}{3}y^2)For (y): (4y - 8y = -4y)Constants: (-3 + 6 = 3)So, the expression becomes:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}y^2 - 4y + 3]Now, complete the square for (y):Factor out (frac{2}{3}):[frac{2}{3}(y^2 - 6y) + 3]Complete the square inside:(y^2 - 6y = (y - 3)^2 - 9)So,[frac{2}{3}[(y - 3)^2 - 9] + 3 = frac{2}{3}(y - 3)^2 - 6 + 3 = frac{2}{3}(y - 3)^2 - 3]Therefore, the entire expression is:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Wait, but the original expression had a +6, and after all this, we have -3. That suggests that the constants have been accounted for correctly. So, the expression is:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Now, since both squared terms are non-negative (because squares are always non-negative), the minimum value occurs when both squares are zero.So, set:1. (x + frac{2}{3}y - 1 = 0)2. (y - 3 = 0)From equation 2, (y = 3).Substitute (y = 3) into equation 1:(x + frac{2}{3}(3) - 1 = 0)Simplify:(x + 2 - 1 = 0 Rightarrow x + 1 = 0 Rightarrow x = -1)So, when (x = -1) and (y = 3), both squared terms are zero, and the expression becomes:[0 + 0 - 3 = -3]Wait, that can't be right because the original expression had a +6, and after completing the squares, we ended up with -3. But when I plug in (x = -1) and (y = 3), let me check the original expression:[3(-1)^2 + 4(-1)(3) + 2(3)^2 - 6(-1) - 8(3) + 6]Calculate each term:- (3(-1)^2 = 3(1) = 3)- (4(-1)(3) = -12)- (2(3)^2 = 2(9) = 18)- (-6(-1) = 6)- (-8(3) = -24)- (+6)Now, sum them up:(3 - 12 + 18 + 6 - 24 + 6)Calculate step by step:3 - 12 = -9-9 + 18 = 99 + 6 = 1515 - 24 = -9-9 + 6 = -3So, yes, the value is indeed -3. But wait, the problem asks for the minimum value, and quadratic expressions can have minima or maxima depending on the coefficients. Since the coefficients of (x^2) and (y^2) are positive, and the quadratic form is positive definite, the expression should have a minimum.But according to my calculation, the minimum value is -3. However, when I look back at the completed square expression:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]The minimum occurs when both squared terms are zero, giving -3. But I need to make sure that this is indeed the minimum.Wait, but let me double-check my completing the square steps because sometimes signs can be tricky.Starting again, the original expression:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]Grouped as:[3x^2 + 4xy - 6x + 2y^2 - 8y + 6]Factored:[3(x^2 + frac{4}{3}xy - 2x) + 2(y^2 - 4y) + 6]Completed the square for (x):[3left[(x + frac{2}{3}y - 1)^2 - (frac{2}{3}y - 1)^2right] + 2(y^2 - 4y) + 6]Expanded the square:[3(x + frac{2}{3}y - 1)^2 - 3(frac{4}{9}y^2 - frac{4}{3}y + 1) + 2y^2 - 8y + 6]Simplified:[3(x + frac{2}{3}y - 1)^2 - frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Combined like terms:[frac{2}{3}y^2 - 4y + 3]Completed the square for (y):[frac{2}{3}(y - 3)^2 - 3]So, the entire expression is:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Yes, that seems correct. So, the minimum value is indeed -3, achieved when both squared terms are zero, i.e., when (x = -1) and (y = 3).But wait, let me check if I didn't make a mistake in the sign when completing the square for (y). Let's go back to that step.After simplifying, we had:[frac{2}{3}y^2 - 4y + 3]Factoring out (frac{2}{3}):[frac{2}{3}(y^2 - 6y) + 3]Completing the square inside:(y^2 - 6y = (y - 3)^2 - 9)So,[frac{2}{3}[(y - 3)^2 - 9] + 3 = frac{2}{3}(y - 3)^2 - 6 + 3 = frac{2}{3}(y - 3)^2 - 3]Yes, that's correct. So, the expression is:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Therefore, the minimum value is -3.But wait, I'm a bit confused because when I plug (x = -1) and (y = 3) into the original expression, I get -3, which seems correct. However, sometimes when completing the square, especially with multiple variables, it's easy to make a mistake. Let me try another approach to verify.Another method to find the minimum of a quadratic function in two variables is to use partial derivatives and set them to zero to find critical points.Let me denote the function as:[f(x, y) = 3x^2 + 4xy + 2y^2 - 6x - 8y + 6]Compute the partial derivatives:[frac{partial f}{partial x} = 6x + 4y - 6][frac{partial f}{partial y} = 4x + 4y - 8]Set both partial derivatives equal to zero:1. (6x + 4y - 6 = 0)2. (4x + 4y - 8 = 0)Now, solve this system of equations.From equation 1:(6x + 4y = 6)Divide both sides by 2:(3x + 2y = 3) --- equation 1aFrom equation 2:(4x + 4y = 8)Divide both sides by 4:(x + y = 2) --- equation 2aNow, solve equations 1a and 2a.From equation 2a: (y = 2 - x)Substitute into equation 1a:(3x + 2(2 - x) = 3)Simplify:(3x + 4 - 2x = 3)Combine like terms:(x + 4 = 3)So, (x = -1)Then, (y = 2 - (-1) = 3)So, the critical point is at (x = -1), (y = 3), which matches the result from completing the square.Now, to confirm that this is indeed a minimum, we can check the second partial derivatives to determine the nature of the critical point.Compute the second partial derivatives:[frac{partial^2 f}{partial x^2} = 6][frac{partial^2 f}{partial y^2} = 4][frac{partial^2 f}{partial x partial y} = 4]The Hessian matrix is:[H = begin{bmatrix}6 & 4 4 & 4 end{bmatrix}]The determinant of the Hessian is:(|H| = (6)(4) - (4)^2 = 24 - 16 = 8)Since the determinant is positive and the second partial derivative with respect to (x) is positive ((6 > 0)), the critical point is a local minimum. Since the function is quadratic and the quadratic form is positive definite (as the Hessian is positive definite), this local minimum is indeed the global minimum.Therefore, the minimum value of the expression is -3, achieved at (x = -1) and (y = 3).Wait, but earlier when I completed the square, I ended up with:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Which suggests that the minimum value is -3. However, when I plug in (x = -1) and (y = 3) into the original expression, I also get -3. So, everything checks out.But just to be thorough, let me try plugging in another point to see if the value is indeed higher. Let's choose (x = 0) and (y = 0):[3(0)^2 + 4(0)(0) + 2(0)^2 - 6(0) - 8(0) + 6 = 6]Which is greater than -3, so that's consistent.Another point: (x = 1), (y = 1):[3(1)^2 + 4(1)(1) + 2(1)^2 - 6(1) - 8(1) + 6 = 3 + 4 + 2 - 6 - 8 + 6 = 1]Again, greater than -3.So, it seems that -3 is indeed the minimum value.Wait, but I'm a bit confused because when I completed the square, I ended up with:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Which is a sum of squares minus 3. Since squares are non-negative, the minimum occurs when both squares are zero, giving -3. That makes sense.Therefore, the minimum value of the expression is (-3), achieved at (x = -1) and (y = 3).But wait, the problem asks for the minimum value, so the answer is (-3). However, sometimes problems like this might have a positive minimum, so I want to make sure I didn't make a mistake in the sign somewhere.Let me double-check the completing the square steps again.Starting with:[3x^2 + 4xy + 2y^2 - 6x - 8y + 6]Grouped as:[3x^2 + 4xy - 6x + 2y^2 - 8y + 6]Factored:[3(x^2 + frac{4}{3}xy - 2x) + 2(y^2 - 4y) + 6]Completed the square for (x):[3left[(x + frac{2}{3}y - 1)^2 - (frac{2}{3}y - 1)^2right] + 2(y^2 - 4y) + 6]Expanded:[3(x + frac{2}{3}y - 1)^2 - 3(frac{4}{9}y^2 - frac{4}{3}y + 1) + 2y^2 - 8y + 6]Simplified:[3(x + frac{2}{3}y - 1)^2 - frac{4}{3}y^2 + 4y - 3 + 2y^2 - 8y + 6]Combined like terms:[frac{2}{3}y^2 - 4y + 3]Completed the square for (y):[frac{2}{3}(y - 3)^2 - 3]So, the entire expression is:[3(x + frac{2}{3}y - 1)^2 + frac{2}{3}(y - 3)^2 - 3]Yes, that's correct. So, the minimum value is indeed -3.Therefore, the minimum value of the expression is (-3).

💡First, I need to determine the percentage of goods lost during theft given that the shopkeeper experiences a 34% loss on the selling price.I'll start by assuming the cost price of the goods is 100. Since the shopkeeper takes a 10% profit, the selling price becomes 110.Next, I'll let x% represent the percentage of goods lost during theft. This means the shopkeeper can only sell (100 - x)% of the goods.The total loss is given as 34%, which means the value of the goods sold is 66% of the original selling price. Calculating 66% of 110 gives me 72.60.The difference between the original selling price and the reduced selling price after theft is 110 - 72.60 = 37.40. This 37.40 represents the value of the x% of goods that were lost.To find the percentage of goods lost, I'll calculate what percentage 37.40 is of the original selling price (110). This gives me (37.40 / 110) * 100 = 34%.Therefore, the shopkeeper lost 34% of his goods during theft.

💡Okay, so I have this equation: (x^2 + y^2 - 12x + 16y + 100 = 0). And I need to find the value of the expression ((x - 7)^{-y}). The options are A: 1, B: -1, C: 8, D: -8. Hmm, let's see how to approach this.First, the equation looks like a circle equation, but it's not in the standard form. Maybe I can rewrite it by completing the square for both (x) and (y). That usually helps in identifying the center and radius of the circle.Starting with the (x) terms: (x^2 - 12x). To complete the square, I take half of -12, which is -6, and square it to get 36. So, I can write (x^2 - 12x) as ((x - 6)^2 - 36).Now for the (y) terms: (y^2 + 16y). Half of 16 is 8, and squaring that gives 64. So, (y^2 + 16y) becomes ((y + 8)^2 - 64).Putting it all back into the original equation:[(x - 6)^2 - 36 + (y + 8)^2 - 64 + 100 = 0]Simplifying the constants: -36 - 64 + 100. Let's calculate that: -36 -64 is -100, and -100 + 100 is 0. So, the equation simplifies to:[(x - 6)^2 + (y + 8)^2 = 0]Wait, the sum of two squares equals zero. Since squares are always non-negative, the only way their sum is zero is if each square is zero individually. So, that means:[x - 6 = 0 quad text{and} quad y + 8 = 0]Solving these gives (x = 6) and (y = -8). Okay, so the only solution is the point (6, -8).Now, I need to compute ((x - 7)^{-y}). Plugging in the values of (x) and (y):[(6 - 7)^{-(-8)} = (-1)^{8}]Wait, because (-y) when (y = -8) is (-(-8) = 8). So, it's ((-1)^8). Any negative number raised to an even power is positive, so ((-1)^8 = 1).So, the value of the expression is 1, which is option A.But just to double-check, let's make sure I didn't make any mistakes in completing the square. For the (x) terms: (x^2 - 12x), completing the square gives ((x - 6)^2 - 36). For the (y) terms: (y^2 + 16y), completing the square gives ((y + 8)^2 - 64). Adding those and the constant term 100, we have:[(x - 6)^2 - 36 + (y + 8)^2 - 64 + 100 = (x - 6)^2 + (y + 8)^2 + (-36 -64 + 100) = (x - 6)^2 + (y + 8)^2 + 0]So yes, it's correct. The equation simplifies to ((x - 6)^2 + (y + 8)^2 = 0), which only holds when both squares are zero, so (x = 6) and (y = -8). Then, substituting into ((x - 7)^{-y}):[(6 - 7)^{-(-8)} = (-1)^8 = 1]Yep, that seems right. So, the answer should be A: 1.**Final Answer**boxed{A}