💡Okay, so I need to figure out how much James pays per episode for his TV show. There are 5 main characters and 4 minor characters. The minor characters each get 15,000 per episode, and the major characters get three times as much as the minor ones. First, I'll start by calculating the total payment for the minor characters. Since there are 4 minor characters and each gets 15,000, I can multiply 4 by 15,000 to get the total for all minor characters. So, 4 times 15,000 is 60,000. That means James pays 60,000 in total to the minor characters each episode.Next, I need to find out how much each major character gets. The problem says that major characters are paid three times as much as the minor ones. Since each minor character gets 15,000, I'll multiply that by 3 to find out the payment per major character.15,000 multiplied by 3 is 45,000. So, each major character receives 45,000 per episode.Now, there are 5 main characters, and each gets 45,000. To find the total payment for all major characters, I'll multiply 5 by 45,000.5 times 45,000 is 225,000. That means James pays 225,000 in total to the major characters each episode.Finally, to find the total amount James pays per episode for all characters, I'll add the total payments for minor and major characters together. So, 60,000 for minor characters plus 225,000 for major characters equals 285,000.Therefore, James pays a total of 285,000 per episode for all the characters.

💡Okay, so I have this problem here with four propositions, labeled A to D, and I need to figure out which one is correct. Let me try to understand each one step by step.Starting with Proposition A: "Two straight lines determine a plane." Hmm, okay. I remember from geometry that if two lines are either intersecting or parallel, they lie on the same plane. So, if I have two straight lines, they can either intersect at a point or be parallel and never meet. In both cases, there's a single plane that contains both lines. So, does that mean two straight lines always determine a plane? Well, I think so, because even if they're parallel, they still lie on the same plane. So, A seems correct.Moving on to Proposition B: "A straight line and a point determine a plane." Alright, so if I have a straight line and a point that's not on that line, can I determine a unique plane? I think yes. There's a principle in geometry that says through a line and a point not on the line, there is exactly one plane. So, if I have a line and a point not on it, they define a unique plane. But wait, what if the point is on the line? Then, infinitely many planes can pass through that line and the point. So, does the proposition specify that the point is not on the line? It just says "a straight line and a point." So, if the point is on the line, it doesn't uniquely determine a plane. Therefore, Proposition B is only correct if the point is not on the line. Since the proposition doesn't specify, it might not always be correct. So, B is conditionally correct but not universally.Next, Proposition C: "If a line is outside a plane, then the line and the plane have no common points." Okay, so if a line is outside a plane, does that mean they don't intersect? Well, if a line is entirely outside a plane, it could either be parallel to the plane or it could intersect the plane at some point. If it's parallel, then they don't intersect, so they have no common points. But if the line is not parallel, it might intersect the plane at one point. So, the proposition says "if a line is outside a plane," which I think means the entire line is not on the plane. But being outside doesn't necessarily mean they don't intersect. They could intersect at a single point. So, Proposition C is not always true because the line could intersect the plane even if it's outside.Finally, Proposition D: "If two lines have no common points, then the two lines are parallel lines or skew lines." Okay, so if two lines don't intersect, what can they be? In three-dimensional space, lines can be parallel, meaning they lie on the same plane and never meet, or they can be skew, meaning they don't lie on the same plane and never meet. So, if two lines have no common points, they must be either parallel or skew. That makes sense. In a plane, two lines that don't intersect are parallel, but in three dimensions, they could also be skew. So, Proposition D seems correct.Wait, but going back to Proposition A, I initially thought it was correct, but let me double-check. If two lines are skew, do they determine a plane? Skew lines are not coplanar, meaning they don't lie on the same plane. So, if two lines are skew, they don't determine a plane because there's no single plane that contains both of them. So, Proposition A says "Two straight lines determine a plane," but if the lines are skew, they don't determine a plane. Therefore, Proposition A is not always correct. It's only correct if the lines are either intersecting or parallel.So, revising my earlier thought, Proposition A is not universally correct because skew lines don't determine a plane. Proposition B is conditionally correct, but not universally because if the point is on the line, it doesn't uniquely determine a plane. Proposition C is not always correct because a line outside a plane could still intersect the plane at a point. Proposition D is correct because if two lines have no common points, they must be either parallel or skew.Therefore, the correct proposition is D.

💡Okay, so I have this problem where I need to find the area of a triangle with vertices at points L(2, 3), M(5, 1), and N(3, 5). Hmm, I remember there are a few ways to find the area of a triangle when you know the coordinates of its vertices. One method is using the Shoelace formula, which I think is also called Gauss's area formula. Another way might be to use vectors or determinants, but I'm not too sure about that. Maybe I'll stick with the Shoelace formula since I think I remember how it works.Alright, so the Shoelace formula says that the area of a polygon, like a triangle in this case, can be found by taking the coordinates of its vertices and plugging them into a specific formula. The formula is:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|Where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three vertices. Let me make sure I got that right. So, I need to assign each point to these variables. Let's say:Point L is (2, 3), so x1 = 2 and y1 = 3.Point M is (5, 1), so x2 = 5 and y2 = 1.Point N is (3, 5), so x3 = 3 and y3 = 5.Okay, now I'll plug these values into the formula.First, calculate each part inside the absolute value:2*(1 - 5) + 5*(5 - 3) + 3*(3 - 1)Let me compute each multiplication step by step.2*(1 - 5) = 2*(-4) = -85*(5 - 3) = 5*(2) = 103*(3 - 1) = 3*(2) = 6Now, add these results together:-8 + 10 + 6Hmm, -8 plus 10 is 2, and then 2 plus 6 is 8.So, the numerator inside the absolute value is 8.Now, divide by 2:8 / 2 = 4And since we're taking the absolute value, it's still 4.So, the area is 4 square units.Wait, does that make sense? Let me double-check my calculations to make sure I didn't make a mistake.First, the coordinates:L(2,3), M(5,1), N(3,5)Shoelace formula:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|Plugging in:|2*(1 - 5) + 5*(5 - 3) + 3*(3 - 1)| / 2Which is |2*(-4) + 5*(2) + 3*(2)| / 2That's |-8 + 10 + 6| / 2Which simplifies to |8| / 2 = 4Okay, that seems correct. But just to be thorough, maybe I can try another method to confirm the area.Another way to find the area of a triangle with coordinates is to use the determinant method, which is similar to the Shoelace formula but presented differently. The formula is:Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Wait, that's essentially the same as the Shoelace formula. So, I guess I'm using the same method again.Alternatively, I could plot the points on a coordinate plane and see if I can calculate the area using base and height. Let's see.Plotting the points:L is at (2,3), M is at (5,1), and N is at (3,5).If I draw these points, I can visualize the triangle. Maybe I can find the lengths of the sides and then use Heron's formula to find the area.First, let's find the lengths of the sides.The distance between L and M:Distance formula: sqrt[(x2 - x1)^2 + (y2 - y1)^2]So, between L(2,3) and M(5,1):sqrt[(5 - 2)^2 + (1 - 3)^2] = sqrt[3^2 + (-2)^2] = sqrt[9 + 4] = sqrt[13]Between M(5,1) and N(3,5):sqrt[(3 - 5)^2 + (5 - 1)^2] = sqrt[(-2)^2 + 4^2] = sqrt[4 + 16] = sqrt[20] = 2*sqrt[5]Between N(3,5) and L(2,3):sqrt[(2 - 3)^2 + (3 - 5)^2] = sqrt[(-1)^2 + (-2)^2] = sqrt[1 + 4] = sqrt[5]So, the lengths of the sides are sqrt(13), 2*sqrt(5), and sqrt(5).Now, using Heron's formula, which requires the semi-perimeter (s) and then the area is sqrt[s(s - a)(s - b)(s - c)].First, calculate the semi-perimeter:s = (sqrt(13) + 2*sqrt(5) + sqrt(5)) / 2 = (sqrt(13) + 3*sqrt(5)) / 2Now, plug into Heron's formula:Area = sqrt[s(s - a)(s - b)(s - c)]But this seems complicated because of the square roots. Maybe it's not the best approach here. Plus, I already got an area of 4 using the Shoelace formula, so unless I made a mistake, this should be correct.Alternatively, I could use vectors to find the area. The area of the triangle is half the magnitude of the cross product of two sides.Let's define vectors from point L to M and from point L to N.Vector LM = M - L = (5 - 2, 1 - 3) = (3, -2)Vector LN = N - L = (3 - 2, 5 - 3) = (1, 2)The cross product of LM and LN is:|LM x LN| = |(3)(2) - (-2)(1)| = |6 + 2| = |8| = 8So, the area is half of that, which is 4.Okay, so using vectors also gives me an area of 4. That matches the Shoelace formula result. So, I'm more confident now that the area is indeed 4 square units.Just to recap, I used two methods: the Shoelace formula and the cross product of vectors, both leading to the same answer. That makes me pretty sure that the area is 4.I think I'm done here. The area of the triangle with vertices at L(2,3), M(5,1), and N(3,5) is 4 square units.

💡Alright, so I need to figure out how to express "110 billion" in scientific notation. Okay, let's start by understanding what scientific notation is. From what I remember, scientific notation is a way to write really big or really small numbers more manageable. It's usually written as a number between 1 and 10 multiplied by 10 raised to some power.So, "110 billion" is a pretty big number. I think "billion" means 10^9, right? So, 110 billion would be 110 times 10^9. But wait, in scientific notation, the number before the 10^something should be between 1 and 10. Right now, 110 is bigger than 10, so I need to adjust it.Let me think, if I have 110 times 10^9, I can write 110 as 11 times 10, right? So, 11 times 10 times 10^9. That would be 11 times 10^(1+9) because when you multiply exponents with the same base, you add the exponents. So, that's 11 times 10^10.But wait, 11 is still bigger than 10. I need to get it down to a number between 1 and 10. How do I do that? Oh, I can write 11 as 1.1 times 10. So, 1.1 times 10 times 10^10. Again, multiplying exponents with the same base, I add the exponents. So, that's 1.1 times 10^(1+10), which is 1.1 times 10^11.Let me double-check that. 10^11 is 100,000,000,000. If I multiply that by 1.1, I get 110,000,000,000, which is 110 billion. Yeah, that makes sense.So, putting it all together, "110 billion" in scientific notation is 1.1 times 10 to the power of 11.

💡First, I need to determine the total payment Jimmy received for delivering pizzas. Jimmy was paid a total of 64.Next, I'll calculate the cost of delivering the 2 pizzas to the building that is 2 kilometers away. Each pizza costs 12, so 2 pizzas cost 2 × 12 = 24. Since the building is more than 1 kilometer away, there is an additional delivery charge of 2. Therefore, the total cost for delivering to the building is 24 + 2 = 26.Now, I'll find out how much Jimmy was paid for delivering pizzas to the park by subtracting the amount paid for the building delivery from the total payment: 64 - 26 = 38.The park is only 100 meters away, which is less than 1 kilometer, so there is no additional delivery charge. This means the entire 38 is for the pizzas themselves. Each pizza costs 12, so the number of pizzas delivered to the park is 38 ÷ 12 = 3 with a remainder of 2. Since partial pizzas aren't counted, Jimmy delivered 3 pizzas to the park.

💡First, I will remove the parentheses from both expressions to make it easier to identify like terms.Next, I will identify and group the like terms together. Like terms are terms that have the same variables raised to the same powers.I will then combine the coefficients of each group of like terms by adding or subtracting them as indicated.After combining all like terms, I will write the simplified expression by listing the terms in descending order of their exponents.Finally, I will present the simplified expression as the final answer.