💡Okay, so I have this problem here about average daily production at a company. Let me try to understand what it's asking. Alright, it says that for the past n days, the average daily production was 50 units. Then, today's production was 95 units, and this raised the average to a certain number. They also mention that n is 8. So, I need to find out what this new average is after including today's production.First, I think I need to figure out the total production for those past n days. Since the average was 50 units per day and n is 8, that should be straightforward. I can multiply the average by the number of days to get the total production.So, total production for 8 days = 50 units/day * 8 days = 400 units.Okay, that makes sense. Now, today's production was 95 units, which is higher than the average. I guess this will increase the overall average. To find the new average, I need to include today's production in the total and then divide by the new number of days.So, the new total production would be the previous total plus today's production: 400 units + 95 units = 495 units.Now, the number of days has increased by one because we're including today. So, the new number of days is 8 + 1 = 9 days.To find the new average, I divide the new total production by the new number of days: 495 units / 9 days.Let me do that division: 495 divided by 9. Hmm, 9 times 50 is 450, and 495 minus 450 is 45. So, 45 divided by 9 is 5. Therefore, 50 + 5 = 55.So, the new average daily production is 55 units per day.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Total production for 8 days: 50 * 8 = 400. Correct.Adding today's production: 400 + 95 = 495. That seems right.Number of days now: 8 + 1 = 9. Makes sense.Dividing total by days: 495 / 9. Let me do this division again. 9 goes into 49 five times (since 9*5=45), subtract 45 from 49, you get 4. Bring down the 5 to make 45. 9 goes into 45 five times. So, it's 55. Yep, that checks out.Alright, so I'm confident now that the new average daily production is 55 units per day.

💡First, I need to determine the total distance the train travels while crossing the bridge. This includes both the length of the train and the length of the bridge.Next, I will calculate the total distance by adding the length of the train (100 meters) to the length of the bridge (140 meters), which gives a total distance of 240 meters.Then, I will use the time taken to cross the bridge (23.998080153587715 seconds) to find the speed of the train in meters per second (m/s) by dividing the total distance by the time.After obtaining the speed in m/s, I will convert it to kilometers per hour (km/h) using the conversion factor of 3.6, since 1 m/s is equivalent to 3.6 km/h.Finally, I will present the calculated speed of the train in kilometers per hour.

💡First, calculate the total cost for the adult tickets by multiplying the number of adults by the cost per adult ticket:9 adults * 11/adult ticket = 99Next, calculate the total cost for the children's tickets by multiplying the number of children by the cost per child ticket:7 children * 7/child ticket = 49Finally, find the difference between the total cost of adult tickets and children's tickets:99 (adults' total) - 49 (children's total) = 50The adults' tickets cost 50 more in total than the children's tickets.

💡First, calculate Robert's total earnings by adding his basic salary and the commission from his sales.Next, determine the amount he allocated to savings by subtracting his monthly expenses from his total earnings.Finally, find the percentage of his total earnings allocated to savings by dividing the savings amount by the total earnings and multiplying by 100.

💡First, I need to determine the total time required for the combined undergrad and Ph.D. program by adding the time for the BS and the Ph.D.Next, since Tom finishes the program in three-fourths of the total time, I will multiply the total time by three-fourths to find out how long it takes him to complete the program.

💡Okay, so I have this function f(x) that's defined in two parts. For x less than or equal to 1, it's a linear function: (a - 2)x - 1. And for x greater than 1, it's a logarithmic function: log base a of x. The question is asking for the range of values for the real number a such that f(x) is monotonically increasing everywhere, from negative infinity to positive infinity.Alright, let's break this down. Monotonically increasing means that as x increases, f(x) should never decrease. So, both pieces of the function need to be increasing on their respective domains, and also, the function needs to be continuous at the point where the pieces meet, which is at x = 1. If it's not continuous there, even if both pieces are increasing, the function could have a jump discontinuity that makes it not increasing overall.First, let's consider the linear part: (a - 2)x - 1 for x ≤ 1. For this to be an increasing function, the slope must be positive. The slope here is (a - 2). So, we need:a - 2 > 0=> a > 2Okay, so a has to be greater than 2 for the linear part to be increasing.Next, the logarithmic part: log base a of x for x > 1. For a logarithmic function log base a of x to be increasing, the base a must be greater than 1. So, we need:a > 1But wait, we already have a > 2 from the linear part, which is a stronger condition. So, the logarithmic part being increasing is already satisfied if a > 2.Now, the crucial part is ensuring that the function is continuous and increasing at the point where the two pieces meet, which is at x = 1. If the function isn't continuous there, or if the right-hand limit is less than the left-hand limit, the function could decrease at that point, which would violate the monotonically increasing condition.So, let's compute the left-hand limit as x approaches 1 from the left and the right-hand limit as x approaches 1 from the right.Left-hand limit (x approaches 1 from the left):lim(x→1⁻) f(x) = (a - 2)(1) - 1 = a - 2 - 1 = a - 3Right-hand limit (x approaches 1 from the right):lim(x→1⁺) f(x) = log_a(1) = 0, since log base a of 1 is always 0.For the function to be continuous at x = 1, these two limits must be equal. So:a - 3 = 0=> a = 3Hmm, so if a = 3, the function is continuous at x = 1. But wait, the problem says f(x) is monotonically increasing on (-∞, +∞). So, does it have to be continuous? Or can it have a jump discontinuity as long as it's still increasing?Let me think. If there's a jump discontinuity at x = 1, then for the function to be monotonically increasing, the right-hand limit must be greater than or equal to the left-hand limit. Otherwise, there would be a decrease at that point.So, more precisely, for f(x) to be increasing, we need:lim(x→1⁺) f(x) ≥ lim(x→1⁻) f(x)Which translates to:0 ≥ a - 3=> a - 3 ≤ 0=> a ≤ 3So, combining this with our earlier condition that a > 2, we get:2 < a ≤ 3Therefore, the range of values for a is (2, 3].Let me double-check this. If a is greater than 2, the linear part is increasing. If a is greater than 1, the logarithmic part is increasing, which is already satisfied since a > 2. At x = 1, the function must not decrease, so the right-hand limit (0) must be greater than or equal to the left-hand limit (a - 3). This gives a ≤ 3. So, putting it all together, a must be greater than 2 and less than or equal to 3.Yes, that makes sense. So, the correct answer should be (2, 3], which is option C.