answer：First, I need to determine the depreciation of the car. John bought the car for 18,000 and traded it in for 6,000. The depreciation is the difference between these two amounts.Next, I calculate the total money John made from driving Uber. This includes both his profit and the depreciation cost. By adding the profit from Uber (18,000) to the depreciation cost (12,000), I can find the total amount he made before considering the car's depreciation.

answer：Okay, so I have this problem here with two propositions, p and q. Let me try to understand what each proposition is saying.First, proposition p: The point (2x + 3 - x², x - 2) is in the fourth quadrant. Hmm, okay. So, in the coordinate plane, the fourth quadrant is where the x-coordinate is positive and the y-coordinate is negative. That means for the point (2x + 3 - x², x - 2) to be in the fourth quadrant, two conditions must be satisfied:1. The x-coordinate must be positive: 2x + 3 - x² > 02. The y-coordinate must be negative: x - 2 < 0So, I need to solve these inequalities to find the range of x for which p is true.Starting with the first inequality: 2x + 3 - x² > 0. Let me rearrange this to make it easier to solve. If I rewrite it as -x² + 2x + 3 > 0, I can multiply both sides by -1 to make the coefficient of x² positive, but I have to remember that multiplying by a negative number reverses the inequality sign. So, that gives me x² - 2x - 3 < 0.Now, let's factor this quadratic. Looking for two numbers that multiply to -3 and add to -2. Hmm, that would be -3 and +1. So, (x - 3)(x + 1) < 0. To solve this inequality, I can use a sign chart. The roots are x = 3 and x = -1. These divide the number line into three intervals:1. x < -1: Test x = -2: (-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 02. -1 < x < 3: Test x = 0: (0 - 3)(0 + 1) = (-3)(1) = -3 < 03. x > 3: Test x = 4: (4 - 3)(4 + 1) = (1)(5) = 5 > 0Since we're looking for where the expression is less than 0, the solution is -1 < x < 3.Now, the second inequality for proposition p is x - 2 < 0, which simplifies to x < 2.So, combining both conditions, the x-values must satisfy both -1 < x < 3 and x < 2. The overlap of these intervals is -1 < x < 2. Therefore, proposition p is true when x is between -1 and 2.Alright, moving on to proposition q: x² - (3a + 6)x + 2a² + 6a < 0. Let me try to factor this quadratic inequality. Let's see if it can be factored.Looking at the quadratic: x² - (3a + 6)x + 2a² + 6a. Let me try to factor it. Let's consider the coefficients:The quadratic is x² - (3a + 6)x + (2a² + 6a). Let me see if I can factor this into two binomials. Let's assume it factors as (x - m)(x - n) where m and n are expressions in terms of a.Expanding (x - m)(x - n) gives x² - (m + n)x + mn. Comparing this to the original quadratic:- (m + n) = -(3a + 6) => m + n = 3a + 6- mn = 2a² + 6aSo, I need two expressions m and n such that their sum is 3a + 6 and their product is 2a² + 6a.Let me try to factor 2a² + 6a. That factors into 2a(a + 3). Hmm, so maybe m = 2a and n = a + 3? Let's check:m + n = 2a + (a + 3) = 3a + 3, which is not equal to 3a + 6. Hmm, close but not quite.Wait, maybe m = a and n = 2a + 6? Let's check:m + n = a + (2a + 6) = 3a + 6. Perfect, that's the sum we need.And mn = a*(2a + 6) = 2a² + 6a. Perfect, that's the product we need.So, the quadratic factors as (x - a)(x - (2a + 6)) < 0.Therefore, the inequality is (x - a)(x - (2a + 6)) < 0.Now, to solve this inequality, I need to consider the roots, which are x = a and x = 2a + 6. The solution will depend on the order of these roots.Case 1: If a = 2a + 6, then the quadratic is a perfect square, and the inequality becomes (x - a)^2 < 0, which is never true because a square is always non-negative. So, in this case, the solution set is empty.Case 2: If a ≠ 2a + 6, which simplifies to a ≠ -6, then we have two distinct roots. The solution to the inequality (x - a)(x - (2a + 6)) < 0 depends on the order of a and 2a + 6.Let's find when a < 2a + 6:a < 2a + 6 => 0 < a + 6 => a > -6.Similarly, if a > 2a + 6, then:a > 2a + 6 => -a > 6 => a < -6.So, if a > -6, then a < 2a + 6, so the roots are ordered as a < 2a + 6. The inequality (x - a)(x - (2a + 6)) < 0 will hold for x between a and 2a + 6.If a < -6, then 2a + 6 < a, so the roots are ordered as 2a + 6 < a. The inequality (x - a)(x - (2a + 6)) < 0 will hold for x between 2a + 6 and a.So, summarizing:- If a > -6: solution is a < x < 2a + 6- If a < -6: solution is 2a + 6 < x < a- If a = -6: no solutionOkay, so that's the solution for q.Now, the problem states that ¬p is a necessary but not sufficient condition for ¬q. Let me unpack that.In logic, if ¬p is a necessary condition for ¬q, it means that whenever ¬q is true, ¬p must also be true. In other words, ¬q implies ¬p. By contrapositive, this is equivalent to p implies q.Additionally, it's stated that ¬p is not a sufficient condition for ¬q. That means that ¬p does not necessarily imply ¬q. In other words, there exists some case where ¬p is true, but ¬q is false (i.e., q is true). So, ¬p does not imply ¬q.So, putting it together:1. ¬q ⇒ ¬p (necessary condition)2. ¬p does not imply ¬q (not sufficient)Which translates to:1. q ⇒ p (contrapositive)2. There exists some x where ¬p is true, but q is true (i.e., ¬p does not necessarily lead to ¬q)Wait, let me think again.If ¬p is a necessary condition for ¬q, then ¬q ⇒ ¬p. So, whenever ¬q is true, ¬p is also true. Equivalently, if q is true, then p must be true (contrapositive). So, q ⇒ p.Also, ¬p is not a sufficient condition for ¬q, meaning that ¬p does not necessarily lead to ¬q. So, there exists some x where ¬p is true, but q is still true.Therefore, in terms of sets:- The solution set of q must be a subset of the solution set of p. Because q ⇒ p, so every x that satisfies q must also satisfy p.- Additionally, the solution set of ¬p must not be entirely contained within the solution set of ¬q. Because ¬p is not a sufficient condition for ¬q, meaning there are some x where ¬p is true, but q is true (so ¬q is false).So, in terms of intervals:- The solution set of q must be a subset of (-1, 2), which is the solution set of p.- The solution set of ¬p is (-∞, -1] ∪ [2, ∞). There must be some overlap between (-∞, -1] ∪ [2, ∞) and the solution set of q. Because if ¬p is not a sufficient condition for ¬q, then there are x in ¬p where q is true.Wait, that might not be the right way to think about it.Wait, let's think in terms of the contrapositive.Since q ⇒ p, the solution set of q must be a subset of p's solution set, which is (-1, 2).Additionally, since ¬p is not a sufficient condition for ¬q, it means that ¬p does not imply ¬q. So, there exists some x where ¬p is true, but q is true. Therefore, the solution set of q must intersect with the solution set of ¬p.But wait, if q is a subset of p, then q cannot intersect with ¬p, because q is entirely within p. So, this seems contradictory.Wait, perhaps I made a mistake in interpreting the necessary and sufficient conditions.Let me recall:If ¬p is a necessary condition for ¬q, then ¬q ⇒ ¬p. So, whenever ¬q is true, ¬p must be true. Equivalently, q ⇒ p.If ¬p is not a sufficient condition for ¬q, then ¬p does not imply ¬q. So, there exists some x where ¬p is true, but ¬q is false (i.e., q is true).So, in terms of sets:- The solution set of q is a subset of p's solution set.- The solution set of q intersects with the solution set of ¬p.But wait, if q is a subset of p, then q cannot intersect with ¬p, because p and ¬p are complements. So, this seems contradictory.Wait, perhaps I need to think differently.Wait, let's re-examine the logic.¬p is a necessary condition for ¬q: So, ¬q ⇒ ¬p. Which is equivalent to p ⇒ q.Wait, no: ¬q ⇒ ¬p is equivalent to p ⇒ q.Wait, no, actually, ¬q ⇒ ¬p is equivalent to p ⇒ q.Wait, let me recall: In logic, the contrapositive of "If A, then B" is "If not B, then not A", and they are logically equivalent.So, if ¬q ⇒ ¬p, then its contrapositive is p ⇒ q. So, p implies q.So, p ⇒ q. So, every x that satisfies p must satisfy q.Additionally, ¬p is not a sufficient condition for ¬q, meaning that ¬p does not necessarily lead to ¬q. So, there exists some x where ¬p is true, but q is true.But wait, if p ⇒ q, then q is a superset of p? Wait, no, p ⇒ q means that p is a subset of q.Wait, no, if p implies q, then whenever p is true, q is true. So, the solution set of p is a subset of the solution set of q.Wait, no, actually, if p ⇒ q, then the solution set of p is a subset of the solution set of q. Because every x that satisfies p must satisfy q.But in our case, p is (-1, 2), and q is an interval depending on a.So, if p ⇒ q, then (-1, 2) must be a subset of the solution set of q.But earlier, we found that the solution set of q is either (a, 2a + 6) if a > -6, or (2a + 6, a) if a < -6.So, for (-1, 2) to be a subset of q's solution set, we need:Case 1: a > -6: q's solution is (a, 2a + 6). So, we need (-1, 2) ⊆ (a, 2a + 6). Therefore:a ≤ -1 and 2a + 6 ≥ 2.Similarly, Case 2: a < -6: q's solution is (2a + 6, a). So, we need (-1, 2) ⊆ (2a + 6, a). But since a < -6, 2a + 6 < a < -6, so the interval (2a + 6, a) is to the left of -6, while (-1, 2) is to the right. So, (-1, 2) cannot be a subset of (2a + 6, a) because they don't overlap. Therefore, Case 2 is impossible.So, only Case 1 is possible: a > -6, and we need:a ≤ -1 and 2a + 6 ≥ 2.Let's solve these inequalities.First, a ≤ -1.Second, 2a + 6 ≥ 2 ⇒ 2a ≥ -4 ⇒ a ≥ -2.Also, since we are in Case 1, a > -6.So, combining these:-2 ≤ a ≤ -1.Additionally, we need to ensure that ¬p is not a sufficient condition for ¬q. That is, there exists some x where ¬p is true, but q is true.Since p ⇒ q, q is a superset of p. So, q includes p and possibly more.But ¬p is the complement of p, which is (-∞, -1] ∪ [2, ∞). For ¬p to not imply ¬q, there must be some x in ¬p where q is true. That is, q must have some overlap with ¬p.But q's solution set is (a, 2a + 6). Since a ≤ -1, let's see:If a ≤ -1, then 2a + 6 ≥ 2*(-1) + 6 = 4. Wait, no, 2a + 6 when a = -1 is 2*(-1) + 6 = 4. But earlier, we have 2a + 6 ≥ 2, which when a = -2, 2*(-2) + 6 = 2. So, the upper bound of q is between 2 and 4.So, the solution set of q is (a, 2a + 6), where a is between -2 and -1, and 2a + 6 is between 2 and 4.Therefore, q's solution set is an interval that starts somewhere between -2 and -1 and ends somewhere between 2 and 4.So, q's solution set is (a, 2a + 6), which includes the interval (-1, 2) as a subset, and extends beyond into (-2, -1] and [2, 4).Therefore, q does overlap with ¬p, specifically in the regions (-2, -1] and [2, 4). So, there are x in ¬p where q is true, meaning that ¬p does not imply ¬q, which satisfies the condition that ¬p is not a sufficient condition for ¬q.Therefore, the range of a is -2 ≤ a ≤ -1.Let me double-check.If a = -2:q's solution is (-2, 2*(-2) + 6) = (-2, 2). So, q is (-2, 2). p is (-1, 2). So, p is a subset of q. Also, q overlaps with ¬p in (-2, -1]. So, ¬p is not sufficient for ¬q.If a = -1:q's solution is (-1, 2*(-1) + 6) = (-1, 4). So, q is (-1, 4). p is (-1, 2). So, p is a subset of q. Also, q overlaps with ¬p in [2, 4). So, ¬p is not sufficient for ¬q.If a is between -2 and -1, say a = -1.5:q's solution is (-1.5, 2*(-1.5) + 6) = (-1.5, 3). So, q is (-1.5, 3). p is (-1, 2). So, p is a subset of q. Also, q overlaps with ¬p in (-1.5, -1] and [2, 3). So, ¬p is not sufficient for ¬q.Therefore, the range of a is indeed -2 ≤ a ≤ -1.

answer：Okay, so I have this equation: 12xy - 18y + 3x - 9/2 = 0, and I need to find the value of x that makes this true for all values of y. Hmm, that means no matter what y is, the equation should hold. So, I guess I need to figure out what x has to be so that this equation doesn't depend on y anymore, or maybe the coefficients of y somehow cancel out.Let me write down the equation again: 12xy - 18y + 3x - 9/2 = 0. I notice that the first two terms have y in them, and the last two terms don't. Maybe I can factor out y from the first two terms. Let's try that.Factoring y out of 12xy - 18y gives me y*(12x - 18). So now the equation looks like y*(12x - 18) + 3x - 9/2 = 0. Okay, so now I have y multiplied by (12x - 18) plus (3x - 9/2) equals zero. Since this equation has to be true for all values of y, that means the coefficient of y must be zero. Otherwise, if y changes, the left side of the equation would change, and it wouldn't equal zero anymore. So, the coefficient of y, which is (12x - 18), must be zero. Let me write that down: 12x - 18 = 0.Solving for x, I can add 18 to both sides: 12x = 18. Then, divide both sides by 12: x = 18/12. Simplifying that fraction, both numerator and denominator are divisible by 6, so 18 divided by 6 is 3, and 12 divided by 6 is 2. So, x = 3/2.But wait, I should also check the constant term to make sure it's zero when x is 3/2. The constant term is 3x - 9/2. Plugging x = 3/2 into that, we get 3*(3/2) - 9/2. Calculating that: 9/2 - 9/2 = 0. Perfect, so the constant term is also zero when x is 3/2.So, putting it all together, when x is 3/2, both the coefficient of y and the constant term become zero, making the entire equation equal to zero for any value of y. That makes sense because if both parts are zero, the equation holds true regardless of y.I think that's the solution. Let me just recap to make sure I didn't miss anything. I factored out y from the terms that had it, set the coefficient of y to zero because the equation must hold for all y, solved for x, and then checked that the constant term also becomes zero with that x. Everything seems to check out. So, x must be 3/2.**Final Answer**The value of ( x ) is boxed{dfrac{3}{2}}.

answer：Alright, so I have this problem here: I need to find the number of integer coordinate points (x, y) such that x² + y² is less than 10. Hmm, okay. Let me think about how to approach this.First off, I know that x and y are integers, so they can be positive, negative, or zero. The condition x² + y² < 10 means that the points (x, y) lie inside a circle centered at the origin with a radius slightly less than √10, which is approximately 3.16. So, the radius is a bit more than 3, but less than 4. That gives me an idea of the range of x and y values I need to consider.Since the radius is about 3.16, the integer values for x and y can range from -3 to 3. That seems manageable. So, I can consider all integer pairs (x, y) where x and y are each in the set {-3, -2, -1, 0, 1, 2, 3}. But not all of these pairs will satisfy x² + y² < 10. I need to find out which ones do.Maybe I can break this down by considering each possible value of x and then determining the possible y values that satisfy the inequality for that x. Let's start with x = 0.When x = 0, the inequality becomes 0 + y² < 10, which simplifies to y² < 10. Since y is an integer, y can be -3, -2, -1, 0, 1, 2, or 3. That's 7 points for x = 0.Next, let's consider x = 1. The inequality becomes 1 + y² < 10, so y² < 9. That means y can be -2, -1, 0, 1, or 2. Wait, no, y² < 9 implies y can be -3, -2, -1, 0, 1, 2, or 3 because 3² is 9, which is not less than 9. Hmm, so actually, y can still be from -3 to 3, but y² must be less than 9. So, y can be -2, -1, 0, 1, or 2. That's 5 points for x = 1.Wait, no, I'm getting confused. If x = 1, then y² < 9, so y can be -2, -1, 0, 1, 2 because (-3)^2 is 9, which is not less than 9. So, y can be -2, -1, 0, 1, 2. That's 5 points.Similarly, for x = -1, the same logic applies because x² is the same whether x is positive or negative. So, for x = -1, y can also be -2, -1, 0, 1, 2. That's another 5 points.Moving on to x = 2. The inequality becomes 4 + y² < 10, so y² < 6. Now, y² < 6 means y can be -2, -1, 0, 1, or 2 because 3² is 9, which is greater than 6. So, y can be -2, -1, 0, 1, 2. That's 5 points for x = 2.Similarly, for x = -2, the same applies. So, another 5 points.Now, x = 3. The inequality becomes 9 + y² < 10, so y² < 1. That means y can only be -0, 0 because 1² is 1, which is not less than 1. So, y can only be 0. That's 1 point for x = 3.Similarly, for x = -3, y can only be 0. So, another 1 point.Wait a second, let me summarize:- x = 0: y = -3, -2, -1, 0, 1, 2, 3 → 7 points- x = 1: y = -2, -1, 0, 1, 2 → 5 points- x = -1: y = -2, -1, 0, 1, 2 → 5 points- x = 2: y = -2, -1, 0, 1, 2 → 5 points- x = -2: y = -2, -1, 0, 1, 2 → 5 points- x = 3: y = 0 → 1 point- x = -3: y = 0 → 1 pointAdding these up: 7 + 5 + 5 + 5 + 5 + 1 + 1 = 29 points.Wait, but I think I might have double-counted some points. Let me check.Actually, no, because for each x, I'm considering all possible y values, and since x ranges from -3 to 3 and y ranges accordingly, each point is unique. So, 29 points in total.But let me double-check by considering symmetry. Since the circle is symmetric in all four quadrants, I can calculate the number of points in one quadrant and multiply by 4, then add the points on the axes.For example, in the first quadrant (x ≥ 0, y ≥ 0):- x = 0: y = 1, 2, 3 → 3 points- x = 1: y = 1, 2 → 2 points- x = 2: y = 1 → 1 point- x = 3: y = 0 → 1 pointSo, in the first quadrant, excluding the axes, we have 3 + 2 + 1 + 1 = 7 points. But wait, actually, when x = 0, y can be 1, 2, 3, which are 3 points. When x = 1, y can be 1, 2, which are 2 points. When x = 2, y can be 1, which is 1 point. When x = 3, y = 0, which is 1 point. So, total in the first quadrant, excluding the axes, is 3 + 2 + 1 + 1 = 7 points.But actually, when x = 0, y = 0 is also a point, but it's on the axis. Similarly, when y = 0, x can be 0, 1, 2, 3, but those are on the axes.So, if I consider all four quadrants, the points not on the axes would be 4 times the number in one quadrant. But I need to be careful not to double-count the axes points.Alternatively, maybe it's easier to just list all the points as I did before.But to ensure I didn't make a mistake, let me list all the points:For x = 0:(0, -3), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (0, 3) → 7 pointsFor x = 1:(1, -2), (1, -1), (1, 0), (1, 1), (1, 2) → 5 points(-1, -2), (-1, -1), (-1, 0), (-1, 1), (-1, 2) → 5 pointsFor x = 2:(2, -2), (2, -1), (2, 0), (2, 1), (2, 2) → 5 points(-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2) → 5 pointsFor x = 3:(3, 0) → 1 point(-3, 0) → 1 pointAdding these up: 7 + 5 + 5 + 5 + 5 + 1 + 1 = 29 points.Yes, that seems correct. So, the total number of integer points (x, y) satisfying x² + y² < 10 is 29.

answer：Okay, so I have this trigonometry problem here. Let me try to figure it out step by step. The problem says:Given that tan(α + π/4) = -1/2, and α is between π/2 and π, I need to find the value of [sin(2α) - 2cos²α] / sin(α - π/4). The options are given as A, B, C, D, so I have to choose the correct one.First, I remember that tan(α + π/4) can be expanded using the tangent addition formula. The formula is tan(A + B) = (tanA + tanB) / (1 - tanA tanB). So, applying that here:tan(α + π/4) = (tanα + tan(π/4)) / (1 - tanα tan(π/4)).I know that tan(π/4) is 1, so substituting that in:tan(α + π/4) = (tanα + 1) / (1 - tanα).According to the problem, this equals -1/2. So, I can set up the equation:(tanα + 1) / (1 - tanα) = -1/2.Now, I need to solve for tanα. Let me cross-multiply to get rid of the fraction:2(tanα + 1) = -1(1 - tanα).Expanding both sides:2tanα + 2 = -1 + tanα.Now, subtract tanα from both sides:2tanα + 2 - tanα = -1.Simplify:tanα + 2 = -1.Subtract 2 from both sides:tanα = -3.Okay, so tanα is -3. Since α is between π/2 and π, which is the second quadrant, I know that in the second quadrant, sine is positive and cosine is negative. So, sinα will be positive, and cosα will be negative.Now, I can think of a right triangle where the opposite side is 3 (since tanα = opposite/adjacent = -3/1, but since we're in the second quadrant, the adjacent side is negative). Wait, actually, tanα is -3, which is negative, so in the second quadrant, the opposite side is positive, and the adjacent side is negative. So, maybe I can represent this as a triangle with opposite side 3 and adjacent side -1.But to find sinα and cosα, I need the hypotenuse. Using the Pythagorean theorem:hypotenuse = sqrt(opposite² + adjacent²) = sqrt(3² + (-1)²) = sqrt(9 + 1) = sqrt(10).So, sinα is opposite/hypotenuse = 3/sqrt(10), and cosα is adjacent/hypotenuse = -1/sqrt(10).But I should rationalize the denominators. So:sinα = 3√10 / 10,cosα = -√10 / 10.Wait, let me check that again. If opposite is 3, adjacent is -1, then hypotenuse is sqrt(10). So, sinα is 3/sqrt(10), which is 3√10/10, and cosα is -1/sqrt(10), which is -√10/10. Yes, that's correct.Now, I need to compute the expression [sin(2α) - 2cos²α] / sin(α - π/4).Let me break this down into parts.First, compute sin(2α). I remember that sin(2α) = 2sinα cosα.So, sin(2α) = 2*(3√10/10)*(-√10/10).Let me compute that:2*(3√10/10)*(-√10/10) = 2*(3*(-1)*10)/(10*10) because √10*√10 is 10.Wait, let me do it step by step.First, multiply 2 with 3√10/10:2*(3√10/10) = (6√10)/10.Then, multiply by (-√10/10):(6√10)/10 * (-√10)/10 = (6*(-1)*(√10*√10))/(10*10).√10*√10 is 10, so:= (6*(-1)*10)/(100) = (-60)/100 = -3/5.So, sin(2α) is -3/5.Next, compute 2cos²α.cosα is -√10/10, so cos²α is ( (-√10/10) )² = (10)/100 = 1/10.So, 2cos²α = 2*(1/10) = 1/5.Therefore, the numerator is sin(2α) - 2cos²α = (-3/5) - (1/5) = (-4/5).Now, the denominator is sin(α - π/4). I can use the sine subtraction formula:sin(A - B) = sinA cosB - cosA sinB.So, sin(α - π/4) = sinα cos(π/4) - cosα sin(π/4).I know that cos(π/4) = sin(π/4) = √2/2.So, substituting the values:sin(α - π/4) = (3√10/10)*(√2/2) - (-√10/10)*(√2/2).Let me compute each term separately.First term: (3√10/10)*(√2/2) = (3√20)/20.Simplify √20: √20 = 2√5, so:= (3*2√5)/20 = (6√5)/20 = (3√5)/10.Second term: (-√10/10)*(√2/2) = (-√20)/20.Again, √20 is 2√5, so:= (-2√5)/20 = (-√5)/10.But since it's subtracting this term, it becomes:sin(α - π/4) = (3√5)/10 - (-√5)/10 = (3√5)/10 + √5/10 = (4√5)/10 = (2√5)/5.So, the denominator is (2√5)/5.Now, putting it all together, the expression is:Numerator: (-4/5)Denominator: (2√5)/5So, (-4/5) divided by (2√5/5) is equal to (-4/5) * (5/(2√5)) = (-4/5)*(5/(2√5)).The 5s cancel out:= (-4)/(2√5) = (-2)/√5.Rationalizing the denominator:= (-2√5)/5.So, the value of the expression is (-2√5)/5, which is option C.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting from tan(α + π/4) = -1/2, I used the tangent addition formula correctly and solved for tanα = -3. Then, since α is in the second quadrant, I correctly identified sinα as positive and cosα as negative, with values 3√10/10 and -√10/10 respectively.Then, computing sin(2α) as 2sinα cosα, I correctly substituted the values and simplified to get -3/5. For 2cos²α, I squared cosα correctly and multiplied by 2 to get 1/5. Subtracting these gives -4/5 for the numerator.For the denominator, using the sine subtraction formula, I correctly expanded sin(α - π/4) and substituted the known values. I correctly simplified each term and combined them to get (2√5)/5.Dividing the numerator (-4/5) by the denominator (2√5/5) gives (-2)/√5, which rationalizes to (-2√5)/5. That matches option C.I think all the steps are correct, so I'm confident that the answer is option C: -2√5/5.**Final Answer**boxed{C}

answer：Alright, so I've got this problem about a square that's divided into four congruent rectangles. Each of these rectangles has a perimeter of 40 inches, and I need to find the perimeter of the square. Hmm, okay, let's break this down step by step.First, I know that a square has all sides equal, so if I can find the length of one side, I can easily find the perimeter by multiplying by four. The square is divided into four congruent rectangles, which means each rectangle is the same size and shape. The Asymptote code draws a square with two lines: one horizontal and one vertical, each splitting the square into halves. So, the square is divided into four smaller rectangles, each occupying a quarter of the square.Let me visualize this. If the square is divided into four equal rectangles, each rectangle must have dimensions that are half the side length of the square in one direction and the full side length in the other. Wait, is that right? If the square is split both horizontally and vertically, then each rectangle would have one side equal to half the square's side and the other side equal to the full side. So, if the square has a side length of 's', then each rectangle has dimensions of (s/2) by s.Okay, so each rectangle has a length of 's' and a width of 's/2'. Now, the perimeter of a rectangle is calculated by the formula: 2*(length + width). So, plugging in the values we have, the perimeter of each rectangle should be 2*(s + s/2). Let me write that out:Perimeter of one rectangle = 2*(s + s/2)Simplify that:= 2*( (2s/2) + (s/2) )= 2*(3s/2)= 3sSo, the perimeter of each rectangle is 3s inches. But the problem states that this perimeter is 40 inches. Therefore, I can set up the equation:3s = 40To find 's', I just need to divide both sides by 3:s = 40 / 3So, the side length of the square is 40/3 inches. Now, to find the perimeter of the square, I multiply the side length by 4:Perimeter of square = 4*s= 4*(40/3)= 160/3Hmm, 160 divided by 3 is approximately 53.333... inches. But since the problem doesn't specify rounding, I should probably leave it as a fraction. So, 160/3 inches is the exact perimeter.Wait, let me double-check my steps to make sure I didn't make a mistake. I assumed the square is divided into four congruent rectangles by splitting it both horizontally and vertically, which makes each rectangle have dimensions of (s/2) by s. Then, using the perimeter formula for a rectangle, I got 3s, set that equal to 40, solved for s, and then found the perimeter of the square. That seems correct.Just to be thorough, let me consider if there's another way the square could be divided into four congruent rectangles. Maybe if the division isn't equal in both directions? But the Asymptote code shows it's split both horizontally and vertically, so each rectangle is indeed a quarter of the square, maintaining the aspect ratio of 1:2. So, my initial assumption holds.Therefore, I'm confident that the perimeter of the square is 160/3 inches.