💡First, calculate the total cost for 5 lines at T-Mobile:- Cost for the first two lines: 50- Cost for each additional line: 16- Number of additional lines needed: 5 - 2 = 3- Total cost at T-Mobile: 50 + (16 × 3) = 98Next, determine the cost for each additional line at M-Mobile:- Let the cost for each additional line at M-Mobile be ( x ) dollars.- Cost for the first two lines at M-Mobile: 45- Total cost at M-Mobile: 45 + (3x)- Since M-Mobile is 11 cheaper than T-Mobile: 98 - (45 + 3x) = 11Solve for ( x ):- 98 - 45 - 3x = 11- 53 - 3x = 11- 53 - 11 = 3x- 42 = 3x- ( x = frac{42}{3} = 14 )Therefore, M-Mobile charges 14 for each additional line.

💡First, I need to determine the speed of the train when it reaches the destination on time. Let's denote this speed as ( v ) km/h.When the train travels at 50 km/h, it is 15 minutes late. Since 15 minutes is a quarter of an hour (0.25 hours), I can set up an equation based on the travel times.The time it takes to travel 15 km at speed ( v ) is ( frac{15}{v} ) hours.The time it takes to travel the same distance at 50 km/h is ( frac{15}{50} ) hours.Since the train is 15 minutes late when traveling at 50 km/h, I can write the equation:[frac{15}{v} + 0.25 = frac{15}{50}]Simplifying the equation:[frac{15}{v} + frac{1}{4} = frac{3}{10}][frac{15}{v} = frac{3}{10} - frac{1}{4}][frac{15}{v} = frac{6}{20} - frac{5}{20}][frac{15}{v} = frac{1}{20}][15 = frac{v}{20}][v = 15 times 20][v = 300]Therefore, the speed of the train when it reaches the destination on time is 300 km/h.

💡Okay, so I'm trying to figure out whether the statement "The appearance of an object observed from different positions is definitely different" is true or false. At first glance, it seems like it might be true because when you move around an object, its appearance does change, right? Like if you have a chair and you walk around it, you see different sides and angles. But wait, the example given earlier was a sphere, which always looks like a circle no matter where you look at it from. So maybe the statement isn't always true.Let me think about this more carefully. If I take a simple shape like a cube, it does look different from different angles. From the front, it's a square, but if I look at it from the top, it's also a square. Hmm, so maybe some objects can look the same from different angles but not all. What about a sphere? No matter where you look at it from, it's always a circle. So in that case, the appearance doesn't change, which means the original statement isn't always true.But what about more complex objects? Like a human face. If you look at someone from the front, you see their whole face, but if you look from the side, you see a profile. So the appearance definitely changes. So it seems like it depends on the object. Some objects change their appearance when viewed from different positions, and some don't.Wait, the question is asking if it's definitely different. So if there's even one object where the appearance doesn't change when viewed from different positions, then the statement is false. The sphere is a good example because it always looks like a circle, no matter the angle. Therefore, the statement isn't universally true.But maybe there's more to it. What about symmetry? Objects with high symmetry, like spheres or cubes, might not change their appearance as much as asymmetrical objects. So the key here is that the statement uses the word "definitely," meaning it must always be different. Since there are objects where the appearance doesn't change, the statement is false.I should also consider if there are any exceptions or special cases. For example, if you have a flat object like a sheet of paper, from the front it's a rectangle, but from the side, it's a line. So the appearance changes. But again, the sphere doesn't change. So it's not that all objects change, but some do and some don't.In summary, the statement is false because there are objects, like spheres, whose appearance remains the same regardless of the viewing position. Therefore, the answer should be false.

💡Alright, so I have this problem about a hotel with room numbers, and I need to figure out how many times the digit 2 is used in all the room numbers. Let me try to break this down step by step.First, the hotel has five floors, numbered from 1 to 5. Each floor has 35 rooms, numbered from n01 to n35, where n is the floor number. So, for example, on the first floor, the rooms are 101, 102, ..., up to 135. Similarly, on the second floor, it's 201, 202, ..., up to 235, and so on up to the fifth floor.The question is asking how many times the digit 2 is used in all these room numbers. So, I need to consider each digit in the three-digit room numbers and count how often a 2 appears.Let me think about the structure of the room numbers. Each room number is a three-digit number where the first digit is the floor number (1-5), and the last two digits are the room number on that floor (01-35). So, for example, room 215 is on the second floor, room number 15.To find how many times the digit 2 is used, I need to consider each digit position separately: the hundreds place (floor number), the tens place, and the units place.1. **Hundreds Place (Floor Number):** - The hundreds digit is the floor number, which ranges from 1 to 5. - So, the digit 2 will appear in the hundreds place only on the second floor. - On the second floor, there are 35 rooms, so the digit 2 will appear 35 times in the hundreds place.2. **Tens Place:** - The tens digit is part of the room number, which ranges from 01 to 35. - I need to count how many times the digit 2 appears in the tens place across all floors. - For each floor, the room numbers go from 01 to 35. So, the tens digit cycles from 0 to 3. - The digit 2 appears in the tens place for room numbers 20 to 29. That's 10 numbers (20, 21, ..., 29). - Since there are five floors, each with 10 occurrences of the digit 2 in the tens place, that's 5 * 10 = 50 times.3. **Units Place:** - Similarly, the units digit cycles from 0 to 9 for each ten's group. - The digit 2 appears in the units place for room numbers ending with 2: 02, 12, 22, 32. - That's 4 occurrences per floor (since room numbers go up to 35). - With five floors, that's 5 * 4 = 20 times.Now, adding up all these occurrences:- Hundreds place: 35 times- Tens place: 50 times- Units place: 20 timesTotal = 35 + 50 + 20 = 105 times.Wait, but let me double-check to make sure I didn't miss anything. For the tens place, on each floor, from 01 to 35, the tens digit is 0,1,2,3. So, 20-29 is indeed 10 numbers, and that's correct for each floor. For the units place, in each floor, the units digit cycles 0-9 for each ten's group, so 02,12,22,32 are the numbers ending with 2, which is 4 per floor. That seems right.And for the hundreds place, only the second floor has 2 in the hundreds place, and there are 35 rooms there, so 35 times. That adds up correctly.So, 35 (hundreds) + 50 (tens) + 20 (units) = 105.I think that's the correct answer. It seems a bit high, but considering that on the second floor, every room number starts with a 2, which contributes 35, and then across all floors, the tens and units places add another 50 and 20, respectively, it does make sense.I don't think I've missed any cases here. Each digit position is considered separately, and the counts are calculated per floor and then summed up. So, I'm confident that the total number of times the digit 2 is used is 105.

💡First, I need to recall the formula for calculating the slope of a line given two points. The slope ( m ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by:[m = frac{y_2 - y_1}{x_2 - x_1}]Next, I'll identify the coordinates of the two points provided in the problem. Point A is ( (1, 2) ) and Point B is ( (3, 6) ).Now, I'll substitute these values into the slope formula:[m = frac{6 - 2}{3 - 1} = frac{4}{2} = 2]Therefore, the slope of the line passing through points A and B is 2.

💡First, I need to understand the given information and what is being asked. We know the percentage of men in the class, the percentage of men who are science majors, and the percentage of non-science majors in the class. The goal is to find out what percentage of the women are science majors.I'll start by assuming the total number of students in the class is 100% for simplicity. This makes the calculations easier.Next, I'll determine the percentage of men who are science majors. Since 40% of the class are men and 70% of these men are science majors, I can calculate this by multiplying 40% by 70%.After finding the percentage of men who are science majors, I'll subtract this from the total percentage of science majors in the class to find out the percentage of women who are science majors.Finally, I'll calculate what percentage of the women this represents by dividing the percentage of women who are science majors by the total percentage of women in the class and then multiplying by 100 to get the final percentage.