💡Okay, so I have this problem where I need to find the intersection of two sets, M and N. Let me read it again carefully to make sure I understand what's being asked.Set M is given as {-2, -1, 0, 1, 2}. That's straightforward. Set N is defined as all real numbers x such that x squared minus x minus 6 is greater than or equal to zero. So, N is the solution set to the inequality x² - x - 6 ≥ 0. I need to find M ∩ N, which means I need to find all elements that are in both M and N.Alright, let's start by solving the inequality for set N. The inequality is x² - x - 6 ≥ 0. To solve this, I think I should factor the quadratic expression if possible. Let me try factoring it.Looking at x² - x - 6, I need two numbers that multiply to -6 and add up to -1. Hmm, let's see. The factors of -6 could be 2 and -3 because 2 * (-3) = -6 and 2 + (-3) = -1. Perfect, so the quadratic factors as (x + 2)(x - 3). So, the inequality becomes (x + 2)(x - 3) ≥ 0.Now, to solve this inequality, I remember that when dealing with quadratic inequalities, it's helpful to find the critical points where the expression equals zero. Those points are x = -2 and x = 3. These points divide the real number line into three intervals: (-∞, -2), (-2, 3), and (3, ∞). I need to test each interval to see where the product (x + 2)(x - 3) is non-negative (greater than or equal to zero).Let's pick a test point in each interval.1. For the interval (-∞, -2), let's choose x = -3. Plugging into (x + 2)(x - 3): (-3 + 2)(-3 - 3) = (-1)(-6) = 6, which is positive. So, this interval satisfies the inequality.2. For the interval (-2, 3), let's choose x = 0. Plugging into (x + 2)(x - 3): (0 + 2)(0 - 3) = (2)(-3) = -6, which is negative. So, this interval does not satisfy the inequality.3. For the interval (3, ∞), let's choose x = 4. Plugging into (x + 2)(x - 3): (4 + 2)(4 - 3) = (6)(1) = 6, which is positive. So, this interval satisfies the inequality.Now, we also need to check the critical points themselves, x = -2 and x = 3, because the inequality is non-strict (≥ 0). At x = -2: ( -2 + 2)( -2 - 3) = (0)(-5) = 0, which satisfies the inequality. Similarly, at x = 3: (3 + 2)(3 - 3) = (5)(0) = 0, which also satisfies the inequality.Therefore, the solution to the inequality x² - x - 6 ≥ 0 is x ≤ -2 or x ≥ 3. So, set N is all real numbers less than or equal to -2 and greater than or equal to 3. In interval notation, that's (-∞, -2] ∪ [3, ∞).Now, set M is {-2, -1, 0, 1, 2}. I need to find the intersection of M and N, which means I need to find the elements that are in both sets.Looking at set M: {-2, -1, 0, 1, 2}Looking at set N: (-∞, -2] ∪ [3, ∞)So, let's check each element of M to see if it's in N.- -2: Is -2 in N? Yes, because N includes all numbers less than or equal to -2. So, -2 is in both M and N.- -1: Is -1 in N? N includes numbers less than or equal to -2 or greater than or equal to 3. -1 is greater than -2, so it's not in N.- 0: Similarly, 0 is greater than -2 and less than 3, so it's not in N.- 1: Same reasoning as above, 1 is not in N.- 2: 2 is also greater than -2 and less than 3, so it's not in N.Therefore, the only element in both M and N is -2. So, the intersection M ∩ N is {-2}.Looking back at the options given:A: {-2, -1, 0, 1}B: {0, 1, 2}C: {-2}D: {2}So, the correct answer should be option C: {-2}.Wait, let me just double-check my work to make sure I didn't make any mistakes.First, solving the inequality: x² - x - 6 ≥ 0.Factored as (x + 2)(x - 3) ≥ 0.Critical points at x = -2 and x = 3.Test intervals:1. x < -2: positive product.2. -2 < x < 3: negative product.3. x > 3: positive product.So, solution is x ≤ -2 or x ≥ 3. That seems correct.Set M is {-2, -1, 0, 1, 2}.Checking each element:- -2 is in N.- -1, 0, 1, 2 are not in N because they are between -2 and 3, which is the interval where the product is negative.Therefore, only -2 is in both sets.Yes, that seems correct. So, the answer is C: {-2}.**Final Answer**boxed{C}

💡First, I recognize that X's current age is given as 45 years.Three years ago, X's age was 45 - 3 = 42 years. According to the problem, this age was double Y's age at that time. Therefore, I can set up the equation:42 = 2 * (Y's age three years ago)Solving for Y's age three years ago:Y's age three years ago = 42 / 2 = 21 yearsNow, to find Y's current age, I add 3 years to Y's age three years ago:Y's current age = 21 + 3 = 24 yearsNext, I calculate their ages seven years from now. Seven years from now, X will be:X's age in seven years = 45 + 7 = 52 yearsAnd Y's age in seven years will be:Y's age in seven years = 24 + 7 = 31 yearsFinally, I find the sum of their ages seven years from now:Sum of their ages = 52 + 31 = 83 years

💡Okay, so I have this trigonometry problem here. It says that sin(x) is equal to -5/13, and x is in the third quadrant. I need to find the absolute value of cos(x/2). Hmm, let me think about how to approach this.First, I remember that in the unit circle, the third quadrant is where both sine and cosine are negative. So, since sin(x) is -5/13, that makes sense for the third quadrant. Now, I need to find cos(x/2). I think I can use the half-angle identity for cosine. Let me recall what that is.The half-angle formula for cosine is:cos(x/2) = ±√[(1 + cos(x))/2]Okay, so I need to find cos(x) first. I know sin(x) is -5/13, and since sin²(x) + cos²(x) = 1, I can solve for cos(x). Let me write that down.sin²(x) + cos²(x) = 1(-5/13)² + cos²(x) = 125/169 + cos²(x) = 1cos²(x) = 1 - 25/169cos²(x) = (169/169 - 25/169)cos²(x) = 144/169So, cos(x) is either 12/13 or -12/13. But since x is in the third quadrant, where cosine is negative, cos(x) must be -12/13.Alright, now I can plug cos(x) into the half-angle formula.cos(x/2) = ±√[(1 + (-12/13))/2]= ±√[(1 - 12/13)/2]= ±√[(1/13)/2]= ±√(1/26)= ±(1/√26)Hmm, 1/√26 can be rationalized as √26/26. So, cos(x/2) is either √26/26 or -√26/26.But the question asks for the absolute value of cos(x/2). So, regardless of the sign, the absolute value is √26/26.Wait, let me double-check. Since x is in the third quadrant, what does that say about x/2? If x is in the third quadrant, then x is between π and 3π/2 radians. So, x/2 would be between π/2 and 3π/4 radians, which is in the second quadrant. In the second quadrant, cosine is negative. So, cos(x/2) should be negative. But since we're taking the absolute value, it doesn't matter. The absolute value is still √26/26.Let me make sure I didn't make any calculation mistakes. Starting from sin(x) = -5/13, found cos(x) = -12/13 correctly. Then applied the half-angle formula, which seems right. The steps look correct, so I think the answer is √26/26.**Final Answer**boxed{frac{sqrt{26}}{26}}

💡To determine the value of ( a ), we start by understanding that the inverse function ( f^{-1}(x) ) passes through the point ( (3, 4) ). This implies that the original function ( f(x) ) passes through the point ( (4, 3) ).Given the function ( f(x) = 1 + log_{a}x ), we substitute ( x = 4 ) and ( f(x) = 3 ) into the equation:[1 + log_{a}4 = 3]Subtracting 1 from both sides:[log_{a}4 = 2]This equation means that ( a ) raised to the power of 2 equals 4:[a^2 = 4]Taking the square root of both sides, we find:[a = 2]Since ( a > 0 ) and ( a neq 1 ), the value of ( a ) is 2.

💡Alright, so I'm trying to figure out how many people need to start the journey on this interstellar vehicle called Gibraltar. It's a huge ship designed to transport families, machinery, and construction materials to distant planets for colonization. The ship can hold 300 family units, each consisting of four members, so that's a total capacity of 1200 people. But the journey will take several years, and families are expected to grow during that time.First, I need to understand the starting population. The problem says that Gibraltar will embark with 100 people less than one-third of its full capacity. So, one-third of 1200 is 400, and subtracting 100 gives us a starting population of 300 people. That seems straightforward.Now, the tricky part is accounting for the growth rates of the families. Families with more than three members grow at a rate of 2% per year, while families with three or fewer members grow at 3% per year. Additionally, the resources on the ship are reduced by 10% each year, which probably affects the population growth somehow.I'm not entirely sure how the resource reduction interacts with the population growth rates. Does it mean that the growth rates are applied to the remaining resources each year? Or does it mean that the population growth is somehow limited by the decreasing resources? Maybe the idea is that as resources decrease, the effective growth rate also decreases.Let me try to break it down step by step. Suppose we start with 300 people. We need to figure out how many families there are and how they're distributed between families with more than three members and those with three or fewer members. But the problem doesn't specify this distribution initially, so maybe I need to assume something about it or find a way to express the total population in terms of these family types.Let's denote:- ( F_{>3} ) as the number of families with more than three members.- ( F_{leq3} ) as the number of families with three or fewer members.Each family contributes a certain number of people to the population. For ( F_{>3} ), let's say each family has, on average, 4 members (since the ship's capacity is based on four-member families). For ( F_{leq3} ), let's assume an average of 3 members.So, the total starting population ( P_0 ) can be expressed as:[ P_0 = 4F_{>3} + 3F_{leq3} ]And we know that ( P_0 = 300 ).Also, the total number of families ( F ) is:[ F = F_{>3} + F_{leq3} ]But we don't know ( F ) initially either.This seems like I need more information to solve for ( F_{>3} ) and ( F_{leq3} ). Maybe I can express the growth in terms of these families and see how the population changes over time.Each year, the number of people in ( F_{>3} ) families grows by 2%, and the number of people in ( F_{leq3} ) families grows by 3%. However, resources are reducing by 10% each year, which might mean that the growth rates are applied to the remaining resources, or perhaps the growth rates are reduced by the resource reduction.I think the key here is to ensure that the population remains stable despite the growth and resource reduction. That means the net growth rate should be zero. So, the growth due to family size increases should be offset by the reduction in resources.Let me try to model this. Suppose after one year, the population would be:[ P_1 = P_0 times (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) ]But resources are reduced by 10%, so maybe:[ P_1 = P_0 times (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ]To maintain stability, ( P_1 = P_0 ), so:[ 1 = (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ]Solving for the growth factor:[ 1 + 0.02 times F_{>3} + 0.03 times F_{leq3} = frac{1}{0.9} approx 1.1111 ][ 0.02 times F_{>3} + 0.03 times F_{leq3} = 0.1111 ]But I also have:[ 4F_{>3} + 3F_{leq3} = 300 ]Now I have two equations:1. ( 4F_{>3} + 3F_{leq3} = 300 )2. ( 0.02F_{>3} + 0.03F_{leq3} = 0.1111 )I can solve this system of equations to find ( F_{>3} ) and ( F_{leq3} ).Let me express ( F_{leq3} ) from the second equation:[ 0.03F_{leq3} = 0.1111 - 0.02F_{>3} ][ F_{leq3} = frac{0.1111 - 0.02F_{>3}}{0.03} ]Now substitute into the first equation:[ 4F_{>3} + 3 times left( frac{0.1111 - 0.02F_{>3}}{0.03} right) = 300 ][ 4F_{>3} + frac{3 times 0.1111 - 0.06F_{>3}}{0.03} = 300 ][ 4F_{>3} + frac{0.3333 - 0.06F_{>3}}{0.03} = 300 ][ 4F_{>3} + 11.1111 - 2F_{>3} = 300 ][ 2F_{>3} + 11.1111 = 300 ][ 2F_{>3} = 288.8889 ][ F_{>3} = 144.4444 ]Since the number of families should be an integer, let's approximate ( F_{>3} approx 144 ).Then,[ F_{leq3} = frac{0.1111 - 0.02 times 144}{0.03} ][ F_{leq3} = frac{0.1111 - 2.88}{0.03} ][ F_{leq3} = frac{-2.7689}{0.03} ][ F_{leq3} approx -92.296 ]Wait, that doesn't make sense. The number of families can't be negative. I must have made a mistake in my calculations.Let me check my steps again. Maybe I misapplied the resource reduction. Perhaps the resource reduction affects the growth rates directly rather than multiplying the entire population.Alternatively, maybe the resource reduction means that the effective growth rate is reduced by 10% each year. So, the growth rates themselves are decreased by 10%.In that case, the growth rates would be:- For ( F_{>3} ): 2% - 10% = -8% (which doesn't make sense because growth can't be negative)- For ( F_{leq3} ): 3% - 10% = -7% (also negative)This also doesn't seem right. Perhaps the resource reduction limits the maximum possible growth. Maybe the growth rates are applied, but the total population can't exceed the remaining resource capacity.This is getting complicated. Maybe I need to approach it differently. Perhaps instead of trying to balance the growth and resource reduction, I should calculate the total population over time considering both factors and find the initial population that results in a stable population.Let me define the population after one year as:[ P_1 = P_0 times (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ]For stability, ( P_1 = P_0 ), so:[ 1 = (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ][ 1 + 0.02 times F_{>3} + 0.03 times F_{leq3} = frac{1}{0.9} approx 1.1111 ][ 0.02 times F_{>3} + 0.03 times F_{leq3} = 0.1111 ]And we still have:[ 4F_{>3} + 3F_{leq3} = 300 ]Let me try solving these equations again more carefully.From the second equation:[ 0.02F_{>3} + 0.03F_{leq3} = 0.1111 ]Let's multiply both sides by 100 to eliminate decimals:[ 2F_{>3} + 3F_{leq3} = 11.11 ]Now, from the first equation:[ 4F_{>3} + 3F_{leq3} = 300 ]Subtract the second equation from the first:[ (4F_{>3} + 3F_{leq3}) - (2F_{>3} + 3F_{leq3}) = 300 - 11.11 ][ 2F_{>3} = 288.89 ][ F_{>3} = 144.445 ]Again, this leads to a non-integer number of families, which isn't practical. Maybe the assumption that the growth rates are applied to the entire population is incorrect. Perhaps the growth rates apply to the number of families, not the population directly.If that's the case, then the number of families increases by 2% or 3% per year, depending on their size. But then, how does that affect the population? Each family's size increases, so the population would grow accordingly.This is getting too confusing. Maybe I need to simplify the problem. Let's assume that all families start with three members, so ( F_{leq3} = 100 ) families (since 100 families x 3 members = 300 people). Then, the growth rate for these families is 3% per year.But resources are reducing by 10% each year, so the effective growth rate would be 3% - 10% = -7%, which doesn't make sense. Maybe the resource reduction limits the growth, so the actual growth rate is 3% of the remaining resources.This is still unclear. Perhaps the key is to realize that to maintain a stable population, the growth must be exactly offset by the resource reduction. So, the net growth rate should be zero.If the growth rate is 3% and resources are reducing by 10%, then the net growth rate is -7%, which would decrease the population. To maintain stability, maybe the initial population needs to account for this net loss.But I'm not sure. I think I need to look for a different approach or maybe re-express the problem in terms of exponential growth and decay.Let me consider the population growth and resource reduction as two opposing forces. The population grows at a certain rate, while resources decrease at another rate. To maintain stability, these two rates must balance each other out.So, if the population grows at a rate ( r ) and resources decrease at a rate ( d ), then for stability:[ r = d ]In this case, the growth rate depends on the family sizes. Families with more than three members grow at 2% per year, and those with three or fewer grow at 3% per year. Resources decrease by 10% per year.So, the overall growth rate ( r ) is a weighted average of 2% and 3%, depending on the proportion of families in each category. The resource decrease rate ( d ) is 10% per year.For stability:[ r = d ][ 0.02F_{>3} + 0.03F_{leq3} = 0.10 ]But we also have:[ 4F_{>3} + 3F_{leq3} = 300 ]Now, we have two equations:1. ( 4F_{>3} + 3F_{leq3} = 300 )2. ( 0.02F_{>3} + 0.03F_{leq3} = 0.10 )Let me solve this system.From equation 2:[ 0.02F_{>3} + 0.03F_{leq3} = 0.10 ]Multiply both sides by 100:[ 2F_{>3} + 3F_{leq3} = 10 ]From equation 1:[ 4F_{>3} + 3F_{leq3} = 300 ]Subtract equation 2 from equation 1:[ (4F_{>3} + 3F_{leq3}) - (2F_{>3} + 3F_{leq3}) = 300 - 10 ][ 2F_{>3} = 290 ][ F_{>3} = 145 ]Then, substitute back into equation 2:[ 2(145) + 3F_{leq3} = 10 ][ 290 + 3F_{leq3} = 10 ][ 3F_{leq3} = -280 ][ F_{leq3} = -93.333 ]Again, a negative number of families doesn't make sense. This suggests that my approach is flawed. Maybe the assumption that the growth rates can balance the resource reduction is incorrect.Perhaps instead of trying to balance them, I need to calculate the total population over time considering both growth and resource reduction and find the initial population that results in a stable population.Let me define the population after one year as:[ P_1 = P_0 times (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ]For stability, ( P_1 = P_0 ), so:[ 1 = (1 + 0.02 times F_{>3} + 0.03 times F_{leq3}) times 0.9 ][ 1 + 0.02 times F_{>3} + 0.03 times F_{leq3} = frac{1}{0.9} approx 1.1111 ][ 0.02 times F_{>3} + 0.03 times F_{leq3} = 0.1111 ]And we still have:[ 4F_{>3} + 3F_{leq3} = 300 ]Let me try solving these equations again.From equation 2:[ 0.02F_{>3} + 0.03F_{leq3} = 0.1111 ]Multiply both sides by 100:[ 2F_{>3} + 3F_{leq3} = 11.11 ]From equation 1:[ 4F_{>3} + 3F_{leq3} = 300 ]Subtract equation 2 from equation 1:[ (4F_{>3} + 3F_{leq3}) - (2F_{>3} + 3F_{leq3}) = 300 - 11.11 ][ 2F_{>3} = 288.89 ][ F_{>3} = 144.445 ]Again, not an integer. Maybe the initial assumption about family sizes is incorrect. Perhaps not all families are at maximum capacity initially.Alternatively, maybe the problem expects a simpler approach, assuming that the population must remain constant despite the growth and resource reduction. In that case, the initial population must already account for the net loss due to resource reduction.If resources are reduced by 10% each year, and assuming the population needs to remain stable, the initial population must be higher to compensate for the loss. However, the problem states that the starting population is 100 less than one-third of capacity, which is 300 people.Given the complexity and the recurring issue of negative family numbers, I think the intended answer might simply be the starting population of 300 people, assuming that the growth rates are nullified by the resource reduction, as initially thought.

💡First, I need to determine the vertical difference between the tops of the two poles. The taller pole is 18 feet tall, and the shorter pole is 8 feet tall. Additionally, the base of the taller pole is 2 feet higher than the base of the shorter pole due to the slope of the ground. This means the total vertical difference between the tops of the poles is (18 - 8 + 2 = 12) feet.Next, I know that the horizontal distance between the bases of the two poles is 20 feet. To find the length of the wire stretched between the tops of the poles, I can use the Pythagorean theorem, which relates the lengths of the sides of a right-angled triangle. The wire forms the hypotenuse of a right triangle where one leg is the horizontal distance (20 feet) and the other leg is the vertical difference (12 feet).Applying the Pythagorean theorem:[c = sqrt{(20)^2 + (12)^2} = sqrt{400 + 144} = sqrt{544}]Therefore, the length of the wire is (sqrt{544}) feet.