💡Okay, so I'm trying to figure out how fast this sports car is moving on the highway. Let me read the problem again and make sure I understand all the details.We have a unidirectional highway, meaning all vehicles are moving in the same direction. There are trucks moving steadily at 60 miles per hour, and they're spaced 1/4 of a mile apart. Then there are cars moving at 75 miles per hour, spaced 3 seconds apart. There's this lone sports car that's weaving through traffic at a steady forward speed, and it passes two cars between each truck it passes. I need to find out how quickly the sports car is moving in miles per hour.Alright, let's break this down. First, I need to understand the spacing between the trucks and the cars. Trucks are 1/4 mile apart, and cars are spaced 3 seconds apart. Since the cars are moving at 75 mph, I can figure out how far apart they are in miles. Similarly, the trucks are moving at 60 mph, so their spacing is given as 1/4 mile.I think it would help to convert the car spacing from seconds to miles. Since the cars are moving at 75 mph, which is miles per hour, I can calculate the distance between each car by multiplying the speed by the time between them. But the time is given in seconds, so I need to convert that to hours.There are 3600 seconds in an hour, so 3 seconds is 3/3600 hours, which simplifies to 1/1200 hours. Now, multiplying the speed of the cars (75 mph) by this time gives me the distance between each car.So, 75 mph * (1/1200) hours = 75/1200 miles. Let me calculate that: 75 divided by 1200 is 0.0625 miles. So, the distance between each car is 1/16 of a mile.Wait, 75 divided by 1200 is indeed 0.0625, which is 1/16. Okay, that makes sense.Now, the trucks are spaced 1/4 mile apart, which is 0.25 miles. So, the distance between trucks is 0.25 miles, and the distance between cars is 0.0625 miles.The sports car is passing two cars between each truck it passes. That means every time the sports car goes from one truck to the next, it passes two cars. So, the distance between two cars is 0.0625 miles, so two cars would be 0.125 miles apart.Wait, no. If the sports car passes two cars between each truck, that means the distance between the two cars that the sports car passes is 0.125 miles. But actually, the cars are moving, so the distance between the cars is 0.0625 miles, but the sports car is moving faster, so it's catching up to the cars.I think I need to consider the relative speed between the sports car and the cars, as well as the relative speed between the sports car and the trucks.Let me denote the speed of the sports car as v mph. The cars are moving at 75 mph, and the trucks are moving at 60 mph. So, the relative speed of the sports car with respect to the cars is (v - 75) mph, and the relative speed with respect to the trucks is (v - 60) mph.Since the sports car passes two cars between each truck, the time it takes to pass two cars should be equal to the time it takes to go from one truck to the next.Let me think about that. The time between passing two cars is the distance between two cars divided by the relative speed between the sports car and the cars. Similarly, the time between passing two trucks is the distance between two trucks divided by the relative speed between the sports car and the trucks.So, the time to pass two cars is (2 * distance between cars) / (relative speed between sports car and cars). The time to pass one truck is (distance between trucks) / (relative speed between sports car and trucks).Since the sports car passes two cars between each truck, these times should be equal.Let me write that down:(2 * distance between cars) / (v - 75) = (distance between trucks) / (v - 60)Plugging in the numbers:2 * (1/16) / (v - 75) = (1/4) / (v - 60)Simplify the left side:2/16 = 1/8, so it's (1/8) / (v - 75)And the right side is (1/4) / (v - 60)So, (1/8) / (v - 75) = (1/4) / (v - 60)Cross-multiplying:(1/8) * (v - 60) = (1/4) * (v - 75)Multiply both sides by 8 to eliminate denominators:(v - 60) = 2 * (v - 75)Expand the right side:v - 60 = 2v - 150Subtract v from both sides:-60 = v - 150Add 150 to both sides:90 = vSo, the speed of the sports car is 90 mph.Wait, let me double-check that.Starting from:(1/8)/(v - 75) = (1/4)/(v - 60)Cross-multiplying:(1/8)(v - 60) = (1/4)(v - 75)Multiply both sides by 8:(v - 60) = 2(v - 75)v - 60 = 2v - 150Subtract v:-60 = v - 150Add 150:90 = vYes, that seems correct.So, the sports car is moving at 90 mph.

💡Alright, so I have this inequality to prove. It looks a bit intimidating at first, but let me try to break it down step by step. The problem states:Given ( a_{1}+a_{2}+cdots+a_{n}=1 ) and ( a_{i} in mathbf{R}_{+} ) for ( i=1,2, cdots, n ), prove that [ frac{a_{1}^{4}}{a_{1}^{3}+a_{1}^{2} a_{2}+a_{1} a_{2}^{2}+a_{2}^{3}}+frac{a_{2}^{4}}{a_{2}^{3}+a_{2}^{2} a_{3}+a_{2} a_{3}^{2}+a_{3}^{3}}+cdots+frac{a_{n}^{4}}{a_{n}^{3}+a_{n}^{2} a_{1}+a_{n} a_{1}^{2}+a_{1}^{3}} geq frac{1}{4}. ]Okay, so we have a sum of fractions, each with a numerator of ( a_i^4 ) and a denominator that's a combination of ( a_i ) and ( a_{i+1} ) terms. The indices wrap around, so ( a_{n+1} = a_1 ). First, I notice that each denominator is a symmetric expression in ( a_i ) and ( a_{i+1} ). Specifically, each denominator is ( a_i^3 + a_i^2 a_{i+1} + a_i a_{i+1}^2 + a_{i+1}^3 ). This looks like a sum of cubes and cross terms. Maybe I can factor this or find a way to simplify it.Let me consider the denominator ( a_i^3 + a_i^2 a_{i+1} + a_i a_{i+1}^2 + a_{i+1}^3 ). Hmm, this seems similar to the expansion of ( (a_i + a_{i+1})^3 ), but not exactly. Let me compute ( (a_i + a_{i+1})^3 ):( (a_i + a_{i+1})^3 = a_i^3 + 3a_i^2 a_{i+1} + 3a_i a_{i+1}^2 + a_{i+1}^3 ).So, the denominator is ( a_i^3 + a_i^2 a_{i+1} + a_i a_{i+1}^2 + a_{i+1}^3 = (a_i + a_{i+1})^3 - 2a_i^2 a_{i+1} - 2a_i a_{i+1}^2 ). Hmm, not sure if that helps.Alternatively, maybe I can factor the denominator. Let me see:( a_i^3 + a_i^2 a_{i+1} + a_i a_{i+1}^2 + a_{i+1}^3 = a_i^3 + a_{i+1}^3 + a_i^2 a_{i+1} + a_i a_{i+1}^2 ).I know that ( a_i^3 + a_{i+1}^3 = (a_i + a_{i+1})(a_i^2 - a_i a_{i+1} + a_{i+1}^2) ), and ( a_i^2 a_{i+1} + a_i a_{i+1}^2 = a_i a_{i+1}(a_i + a_{i+1}) ). So, combining these:Denominator = ( (a_i + a_{i+1})(a_i^2 - a_i a_{i+1} + a_{i+1}^2) + a_i a_{i+1}(a_i + a_{i+1}) ).Factor out ( (a_i + a_{i+1}) ):Denominator = ( (a_i + a_{i+1})(a_i^2 - a_i a_{i+1} + a_{i+1}^2 + a_i a_{i+1}) ).Simplify inside the parentheses:( a_i^2 - a_i a_{i+1} + a_{i+1}^2 + a_i a_{i+1} = a_i^2 + a_{i+1}^2 ).So, denominator = ( (a_i + a_{i+1})(a_i^2 + a_{i+1}^2) ).Ah, that's a nice simplification! So, each denominator can be written as ( (a_i + a_{i+1})(a_i^2 + a_{i+1}^2) ).So, the fraction becomes:( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Now, let me consider the entire sum:( sum_{i=1}^n frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).I need to show that this sum is at least ( frac{1}{4} ).Hmm, perhaps I can use some inequality here. Maybe Cauchy-Schwarz or AM-GM?Let me think about Cauchy-Schwarz. The Cauchy-Schwarz inequality states that for sequences ( (x_i) ) and ( (y_i) ), we have:( (sum x_i y_i)^2 leq (sum x_i^2)(sum y_i^2) ).But I'm not sure how to apply it directly here. Alternatively, maybe I can manipulate each term.Looking at each term ( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ), perhaps I can write it as:( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} = frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Wait, maybe I can split the fraction:( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} = frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Alternatively, perhaps I can write ( a_i^4 = a_i^2 cdot a_i^2 ), so:( frac{a_i^2 cdot a_i^2}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Hmm, maybe I can use the AM-GM inequality on the denominator. The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is at least the geometric mean.But I'm not sure yet. Let me think differently. Maybe I can consider the entire sum and see if I can bound it below.Alternatively, perhaps I can use the Cauchy-Schwarz inequality in the following way. For each term, consider:( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} geq frac{a_i^4}{(a_i + a_{i+1}) cdot frac{(a_i^2 + a_{i+1}^2)}{2} cdot 2} ).Wait, that might not be helpful. Alternatively, perhaps I can use the fact that ( a_i^2 + a_{i+1}^2 geq frac{(a_i + a_{i+1})^2}{2} ) by the Cauchy-Schwarz inequality.Yes, that's a useful inequality. Specifically, ( a_i^2 + a_{i+1}^2 geq frac{(a_i + a_{i+1})^2}{2} ).So, substituting this into the denominator, we get:( (a_i + a_{i+1})(a_i^2 + a_{i+1}^2) geq (a_i + a_{i+1}) cdot frac{(a_i + a_{i+1})^2}{2} = frac{(a_i + a_{i+1})^3}{2} ).Therefore, each term satisfies:( frac{a_i^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} geq frac{a_i^4}{frac{(a_i + a_{i+1})^3}{2}} = frac{2a_i^4}{(a_i + a_{i+1})^3} ).So, the entire sum ( M ) satisfies:( M geq sum_{i=1}^n frac{2a_i^4}{(a_i + a_{i+1})^3} ).Hmm, but I'm not sure if this helps me directly. Maybe I need a different approach.Let me consider the sum ( M ) and another similar sum ( N ) where each term is ( frac{a_{i+1}^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ). So, ( N = sum_{i=1}^n frac{a_{i+1}^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Then, if I consider ( M - N ), I get:( M - N = sum_{i=1}^n left( frac{a_i^4 - a_{i+1}^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} right) ).I can factor ( a_i^4 - a_{i+1}^4 ) as ( (a_i - a_{i+1})(a_i + a_{i+1})(a_i^2 + a_{i+1}^2) ). So, substituting this in:( M - N = sum_{i=1}^n frac{(a_i - a_{i+1})(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} ).Simplifying, the ( (a_i + a_{i+1}) ) and ( (a_i^2 + a_{i+1}^2) ) terms cancel out, leaving:( M - N = sum_{i=1}^n (a_i - a_{i+1}) ).But this sum telescopes:( sum_{i=1}^n (a_i - a_{i+1}) = a_1 - a_2 + a_2 - a_3 + cdots + a_n - a_1 = 0 ).So, ( M - N = 0 ), which implies ( M = N ).Now, since ( M = N ), I can consider ( M + N = 2M ). Let's compute ( M + N ):( M + N = sum_{i=1}^n left( frac{a_i^4 + a_{i+1}^4}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} right) ).Now, I can apply the AM-GM inequality to the numerator ( a_i^4 + a_{i+1}^4 ). The AM-GM inequality states that ( frac{a_i^4 + a_{i+1}^4}{2} geq sqrt{a_i^4 a_{i+1}^4} = a_i^2 a_{i+1}^2 ). But I'm not sure if that's directly helpful.Alternatively, perhaps I can use the Cauchy-Schwarz inequality in the form:( (a_i^4 + a_{i+1}^4)(1 + 1) geq (a_i^2 + a_{i+1}^2)^2 ).This is because ( (a_i^4 + a_{i+1}^4)(1 + 1) geq (a_i^2 + a_{i+1}^2)^2 ) by Cauchy-Schwarz.So, ( a_i^4 + a_{i+1}^4 geq frac{(a_i^2 + a_{i+1}^2)^2}{2} ).Substituting this into ( M + N ):( M + N geq sum_{i=1}^n frac{frac{(a_i^2 + a_{i+1}^2)^2}{2}}{(a_i + a_{i+1})(a_i^2 + a_{i+1}^2)} = sum_{i=1}^n frac{(a_i^2 + a_{i+1}^2)}{2(a_i + a_{i+1})} ).Simplifying further:( M + N geq frac{1}{2} sum_{i=1}^n frac{a_i^2 + a_{i+1}^2}{a_i + a_{i+1}} ).Now, I can apply the AM-GM inequality again to each term ( frac{a_i^2 + a_{i+1}^2}{a_i + a_{i+1}} ). Specifically, ( frac{a_i^2 + a_{i+1}^2}{a_i + a_{i+1}} geq frac{a_i + a_{i+1}}{2} ). This is because ( a_i^2 + a_{i+1}^2 geq frac{(a_i + a_{i+1})^2}{2} ), so dividing both sides by ( a_i + a_{i+1} ) gives ( frac{a_i^2 + a_{i+1}^2}{a_i + a_{i+1}} geq frac{a_i + a_{i+1}}{2} ).Therefore:( M + N geq frac{1}{2} sum_{i=1}^n frac{a_i + a_{i+1}}{2} = frac{1}{4} sum_{i=1}^n (a_i + a_{i+1}) ).But ( sum_{i=1}^n (a_i + a_{i+1}) = 2(a_1 + a_2 + cdots + a_n) = 2 times 1 = 2 ), since ( a_1 + a_2 + cdots + a_n = 1 ).Thus:( M + N geq frac{1}{4} times 2 = frac{1}{2} ).Since ( M = N ), we have ( 2M geq frac{1}{2} ), which implies ( M geq frac{1}{4} ).Therefore, the original inequality is proven.

💡Alright, let's tackle this problem step by step. It's about figuring out the percentage of employed females in town X after five years, considering the growth in employment rates and the introduction of a new employment category. Plus, we need to determine which category the majority of these females work in. Okay, let's break it down.First, we know that currently, 64% of the population in town X are employed. Out of these employed individuals, 55% are males. So, the first thing I want to find out is what percentage of the employed population are females. That should be straightforward.So, if 55% are males, then the remaining percentage must be females. That would be 100% minus 55%, which is 45%. So, currently, 45% of the employed population are females.Now, the problem mentions that over the past five years, the employment rate has been increasing by 2% annually. The initial employment rate was 64%. So, we need to calculate the employment rate after five years. To do this, I think we can use the formula for compound growth, which is:Final Amount = Initial Amount × (1 + Growth Rate)^Number of PeriodsIn this case, the initial amount is 64%, the growth rate is 2% per year, and the number of periods is 5 years.So, plugging in the numbers:Final Employment Rate = 64% × (1 + 0.02)^5Let me calculate that. First, (1 + 0.02) is 1.02. Raising that to the power of 5:1.02^5 ≈ 1.10408So, Final Employment Rate ≈ 64% × 1.10408 ≈ 70.74%So, after five years, the employment rate is approximately 70.74%.But wait, the problem also mentions that in the last year, a new employment category, Tourism, was introduced, accounting for 10% of the total employed population, with an equal proportion of male and female employees.Hmm, so in the fifth year, Tourism accounts for 10% of the total employed population. Since the proportion of male and female employees in Tourism is equal, this means that 5% of the employed population in Tourism are males and 5% are females.But does this affect the overall percentage of employed females? Let's think about it.Originally, before the introduction of Tourism, 45% of the employed population were females. Now, with Tourism adding 10% to the total employed population, and 5% of that being females, we need to adjust the total percentage of employed females.Wait, actually, the total employed population has increased by 2% each year for five years, so the total employed population is now approximately 70.74%. Within this, 10% is Tourism, which is 10% of 70.74%, which is about 7.074%.Out of this 7.074%, 5% are females, so that's about 0.3537% of the total population.But this seems a bit confusing. Maybe I need to approach it differently.Perhaps, instead of adjusting the percentages directly, I should consider the composition of the employed population after the introduction of Tourism.Originally, 45% of the employed population were females. Now, with Tourism adding 10% to the total employed population, and 5% of that being females, the total percentage of females in the employed population becomes:Original females: 45%Females in Tourism: 5% of 10% = 0.5%So, total females: 45% + 0.5% = 45.5%Wait, that doesn't seem right. Because the total employed population has increased, so the percentages need to be adjusted accordingly.Let me think again.The initial employed population is 64%. After five years of 2% growth, it's approximately 70.74%. Within this, 10% is Tourism, which is 7.074% of the total population. Of this 7.074%, 5% are females, so that's 0.3537% of the total population.But the original 64% employed population had 45% females, which is 28.8% of the total population. Adding the 0.3537% from Tourism, the total females employed would be 28.8% + 0.3537% ≈ 29.1537% of the total population.But the total employed population is now 70.74%, so the percentage of females in the employed population is:(29.1537% / 70.74%) × 100 ≈ 41.2%Wait, that's lower than the original 45%. That doesn't make sense because we added females in Tourism. Maybe I made a mistake.Alternatively, perhaps the 10% Tourism is of the total employed population after growth, which is 70.74%. So, 10% of 70.74% is 7.074%. Of this, 5% are females, so 0.3537% of the total population.But the original 64% employed population had 45% females, which is 28.8% of the total population. Adding 0.3537% gives 29.1537% of the total population as females employed.But the total employed population is now 70.74%, so the percentage of females in the employed population is:(29.1537% / 70.74%) × 100 ≈ 41.2%Hmm, so the percentage of females in the employed population decreased from 45% to approximately 41.2% after the introduction of Tourism.But that seems counterintuitive because we added females in Tourism. Maybe I need to consider that the original 64% is the base, and the growth is on top of that.Wait, perhaps the growth rate is applied to the initial 64%, so each year it's 64% × 1.02, then the next year it's that result × 1.02, and so on for five years.Let me recalculate the final employment rate:Year 1: 64% × 1.02 = 65.28%Year 2: 65.28% × 1.02 ≈ 66.5856%Year 3: 66.5856% × 1.02 ≈ 67.9173%Year 4: 67.9173% × 1.02 ≈ 69.2757%Year 5: 69.2757% × 1.02 ≈ 70.6512%So, approximately 70.65% after five years.Now, within this 70.65%, 10% is Tourism, which is 7.065% of the total population. Of this, 5% are females, so 0.35325% of the total population.The original 64% employed population had 45% females, which is 28.8% of the total population. Adding 0.35325% gives 29.15325% of the total population as females employed.But the total employed population is now 70.65%, so the percentage of females in the employed population is:(29.15325% / 70.65%) × 100 ≈ 41.26%So, approximately 41.26% of the employed population are females after five years.But the problem asks for the percentage of employed people who are females, not the percentage of the total population. So, I think the correct approach is to consider the composition of the employed population.Originally, 55% of the total population are employed males, so 45% are employed females. After five years, the total employed population is 70.65% of the total population. Within this, 10% is Tourism, which is 7.065% of the total population. Of this, 5% are females, so 0.35325% of the total population.But the original 64% employed population had 45% females, which is 28.8% of the total population. Adding 0.35325% gives 29.15325% of the total population as females employed.But the total employed population is now 70.65%, so the percentage of females in the employed population is:(29.15325% / 70.65%) × 100 ≈ 41.26%So, approximately 41.26% of the employed population are females after five years.But wait, the problem also mentions that female employees are only in Health and Education, with 60% of them in Education. So, the majority of employed females work in Education.Therefore, the final answer is that approximately 41.26% of the employed people in town X are females in the fifth year, and the majority of them work in the Education category.But the problem asks for the percentage, so I think we can round it to 41.3%.However, the initial calculation without considering the growth rate gave us 45%, and after considering the growth and the introduction of Tourism, it decreased to approximately 41.3%.But I'm not sure if this is the correct approach. Maybe I should consider the growth rate differently.Alternatively, perhaps the growth rate affects the number of employed people, but the percentage of males and females remains the same unless affected by the new category.So, if the employment rate increases by 2% each year, the total number of employed people increases, but the percentage of males and females among them remains the same unless the new category changes it.In the last year, Tourism was introduced, accounting for 10% of the total employed population, with an equal proportion of male and female employees. So, in the fifth year, 10% of the employed population is in Tourism, with 5% males and 5% females.Therefore, the total percentage of females in the employed population would be the original 45% plus 5% from Tourism, but since the total employed population has increased, we need to adjust accordingly.Wait, perhaps it's better to calculate the total number of employed people after five years and then see how many are females.Let's assume the total population is 100 for simplicity.Initially, 64% are employed, so 64 people are employed. Out of these, 55% are males, so 35.2 males and 28.8 females.Over five years, the employment rate increases by 2% each year. So, the employed population grows as follows:Year 1: 64 × 1.02 = 65.28Year 2: 65.28 × 1.02 ≈ 66.5856Year 3: 66.5856 × 1.02 ≈ 67.9173Year 4: 67.9173 × 1.02 ≈ 69.2757Year 5: 69.2757 × 1.02 ≈ 70.6512So, after five years, approximately 70.6512 people are employed.In the fifth year, Tourism accounts for 10% of the total employed population, which is 10% of 70.6512 ≈ 7.0651 people. Of these, 5% are females, so approximately 0.35325 females.But wait, the 10% is of the total employed population, which is 70.6512, so 10% is 7.0651 people. Of these, 50% are females, so 3.53255 females.Wait, I think I made a mistake earlier. The 10% is of the total employed population, which is 70.6512, so 10% is 7.0651 people. Of these, 50% are females, so 3.53255 females.So, the total number of females employed is the original 28.8 plus 3.53255 ≈ 32.33255.But wait, the original 64 employed people had 28.8 females. After five years, the total employed is 70.6512, so the number of females is 28.8 plus 3.53255 ≈ 32.33255.Therefore, the percentage of females in the employed population is:(32.33255 / 70.6512) × 100 ≈ 45.75%Wait, that's higher than the original 45%. That seems contradictory to my earlier calculation.I think the confusion arises from whether the 10% Tourism is added to the total employed population or if it's part of the growth.Let me clarify:The employment rate increases by 2% annually for five years, leading to a total employed population of approximately 70.65% of the total population.In the fifth year, Tourism is introduced, accounting for 10% of the total employed population. So, 10% of 70.65% is 7.065% of the total population.Of this 7.065%, 50% are females, so 3.5325% of the total population are females in Tourism.The original 64% employed population had 45% females, which is 28.8% of the total population.Adding the 3.5325% from Tourism, the total females employed are 28.8% + 3.5325% ≈ 32.3325% of the total population.But the total employed population is now 70.65%, so the percentage of females in the employed population is:(32.3325% / 70.65%) × 100 ≈ 45.75%So, approximately 45.75% of the employed population are females after five years.This makes sense because we added females in Tourism, which increased the percentage slightly.But wait, the original percentage was 45%, and after adding females in Tourism, it increased to approximately 45.75%. So, the percentage increased slightly.But the problem asks for the percentage after considering growth rates and the introduction of the new employment category. So, the final percentage is approximately 45.75%.But the problem also mentions that female employees are only in Health and Education, with 60% of them in Education. So, the majority of employed females work in Education.Therefore, the final answer is that approximately 45.75% of the employed people in town X are females in the fifth year, and the majority of them work in the Education category.But the problem asks for the percentage, so I think we can round it to 46%.However, to be precise, let's do the exact calculation:Total females employed = 28.8 + 3.5325 = 32.3325Total employed population = 70.6512Percentage of females = (32.3325 / 70.6512) × 100 ≈ 45.75%So, approximately 45.75%, which is about 46%.Therefore, the percentage of employed females in the fifth year is approximately 46%, and the majority of them work in Education.I think this makes sense now. The growth in employment rate increased the total number of employed people, and adding females in Tourism slightly increased the percentage of females in the employed population.

💡Okay, so I have this problem here: Given the equation ( x + 3y - 1 = 0 ), I need to figure out something about the expression ( 2^x + 8^y ). The options are about whether it has a maximum or minimum value, and the specific values given are 8, ( 2sqrt{2} ), etc. Hmm, let me try to break this down step by step.First, the equation given is ( x + 3y - 1 = 0 ). That simplifies to ( x + 3y = 1 ). So, this is a linear equation relating x and y. I can probably express one variable in terms of the other. Let me solve for x: ( x = 1 - 3y ). Alternatively, I could solve for y: ( y = frac{1 - x}{3} ). Not sure which one will be more useful yet, but it's good to have both options.Now, the expression I need to analyze is ( 2^x + 8^y ). Hmm, both terms are exponential functions with base 2. Wait, 8 is ( 2^3 ), so maybe I can rewrite 8^y in terms of base 2. Let me try that: ( 8^y = (2^3)^y = 2^{3y} ). So, the expression becomes ( 2^x + 2^{3y} ). That's interesting because both terms now have the same base, which might make it easier to apply some kind of inequality or optimization technique.Since I have ( x + 3y = 1 ), and I have ( 2^x + 2^{3y} ), maybe I can express both exponents in terms of a single variable. Let me substitute x from the equation into the expression. If ( x = 1 - 3y ), then ( 2^x = 2^{1 - 3y} ). So, the expression becomes ( 2^{1 - 3y} + 2^{3y} ). Hmm, that's ( 2 cdot 2^{-3y} + 2^{3y} ). Let me write that as ( 2 cdot (2^{-3y}) + 2^{3y} ).Wait, that looks like something I can maybe apply the AM-GM inequality to. The AM-GM inequality states that for non-negative real numbers a and b, the arithmetic mean is greater than or equal to the geometric mean, so ( frac{a + b}{2} geq sqrt{ab} ). Equality holds when a = b. Let me see if I can apply that here.Let me denote ( a = 2 cdot 2^{-3y} ) and ( b = 2^{3y} ). Then, the arithmetic mean would be ( frac{a + b}{2} = frac{2 cdot 2^{-3y} + 2^{3y}}{2} ), and the geometric mean would be ( sqrt{a cdot b} = sqrt{2 cdot 2^{-3y} cdot 2^{3y}} ). Let me compute that.First, the geometric mean: ( sqrt{2 cdot 2^{-3y} cdot 2^{3y}} ). The exponents on 2 are -3y and 3y, so when multiplied, they add up: ( -3y + 3y = 0 ). So, that simplifies to ( sqrt{2 cdot 2^0} = sqrt{2 cdot 1} = sqrt{2} ). So, the geometric mean is ( sqrt{2} ).Therefore, by AM-GM, ( frac{a + b}{2} geq sqrt{2} ), which implies ( a + b geq 2sqrt{2} ). So, ( 2 cdot 2^{-3y} + 2^{3y} geq 2sqrt{2} ). That means the expression ( 2^x + 8^y ) has a minimum value of ( 2sqrt{2} ).Wait, but let me double-check. When does equality hold in AM-GM? It holds when a = b. So, in this case, when ( 2 cdot 2^{-3y} = 2^{3y} ). Let me solve for y.( 2 cdot 2^{-3y} = 2^{3y} )Divide both sides by ( 2^{-3y} ):( 2 = 2^{3y} / 2^{-3y} = 2^{3y + 3y} = 2^{6y} )So, ( 2 = 2^{6y} ), which implies ( 6y = 1 ), so ( y = frac{1}{6} ).Then, from the equation ( x + 3y = 1 ), plugging in y = 1/6:( x + 3*(1/6) = 1 ) => ( x + 1/2 = 1 ) => ( x = 1/2 ).So, when x = 1/2 and y = 1/6, the expression ( 2^x + 8^y ) equals ( 2sqrt{2} ). Therefore, this is indeed the minimum value.Now, looking back at the options:A: It has a maximum value of 8.B: It has a minimum value of ( 2sqrt{2} ).C: It has a minimum value of 8.D: It has a maximum value of ( 2sqrt{2} ).From my analysis, the expression has a minimum value of ( 2sqrt{2} ), so option B is correct. But just to be thorough, let me check if there's a maximum value or if the expression can go to infinity.Since both ( 2^x ) and ( 8^y ) are exponential functions, as x or y increase, the expression can become arbitrarily large. However, since x and y are related by ( x + 3y = 1 ), if x increases, y must decrease, and vice versa. Let me see what happens as y approaches infinity or negative infinity.Wait, if y approaches infinity, then from ( x = 1 - 3y ), x would approach negative infinity. So, ( 2^x ) would approach 0, but ( 8^y = 2^{3y} ) would approach infinity. Similarly, if y approaches negative infinity, then x approaches positive infinity, so ( 2^x ) approaches infinity, while ( 8^y ) approaches 0. So, in both cases, the expression ( 2^x + 8^y ) approaches infinity. Therefore, there is no maximum value; it can grow without bound.Thus, the only correct statement is that it has a minimum value of ( 2sqrt{2} ), which is option B.

💡To find the distance between the two parallel lines ( l_1: x - y + 1 = 0 ) and ( l_2: x - y + 3 = 0 ), we can use the formula for the distance between two parallel lines. The general formula for the distance ( d ) between two parallel lines ( Ax + By + C_1 = 0 ) and ( Ax + By + C_2 = 0 ) is:[d = frac{|C_2 - C_1|}{sqrt{A^2 + B^2}}]For the given lines:- ( A = 1 )- ( B = -1 )- ( C_1 = 1 )- ( C_2 = 3 )Plugging these values into the formula:[d = frac{|3 - 1|}{sqrt{1^2 + (-1)^2}} = frac{2}{sqrt{2}} = sqrt{2}]Therefore, the distance between the two parallel lines is (boxed{sqrt{2}}).

💡Okay, so I've got this problem here: If for any a and b where 0 < a < b < t, it holds that b ln a < a ln b, then what's the maximum value of t? And the answer is supposed to be e. Hmm, interesting. I need to figure out why that is.Alright, let's start by understanding the inequality given: b ln a < a ln b. Since a and b are positive numbers with a < b, and both are less than t. I need to find the maximum t such that this inequality always holds for any a and b in that interval.Maybe I can rearrange the inequality to make it more manageable. Let's see, if I divide both sides by a b, which are positive, so the inequality sign won't change. That gives me (ln a)/a < (ln b)/b. So, the function f(x) = (ln x)/x is increasing? Because if a < b, then f(a) < f(b). So, f is increasing on the interval (0, t). Wait, so if f(x) is increasing on (0, t), then t must be such that f(x) is increasing up to t. So, I need to find the maximum t where f(x) is increasing. To find where f(x) is increasing, I can take its derivative and see where it's positive.Let me compute the derivative of f(x) = (ln x)/x. Using the quotient rule: f'(x) = [ (1/x)*x - ln x *1 ] / x². Simplifying that, it's [1 - ln x] / x². So, f'(x) = (1 - ln x)/x².Now, for f'(x) to be positive, the numerator must be positive because the denominator x² is always positive for x ≠ 0. So, 1 - ln x > 0. That implies ln x < 1, which means x < e^1, so x < e.Therefore, f(x) is increasing on the interval (0, e). So, if t is greater than e, then f(x) would start decreasing beyond e, right? So, if t were larger than e, say t = e + 1, then for some a and b where e < a < b < t, f(a) might not be less than f(b) because f(x) is decreasing there. That would violate the given condition.Hence, the maximum value of t where f(x) is increasing is e. So, t can't be larger than e because beyond e, the function starts decreasing, and the inequality b ln a < a ln b might not hold anymore.Wait, let me test this with some numbers to make sure. Let's take t = e, and pick a = 1 and b = e. Then, b ln a = e ln 1 = 0, and a ln b = 1 ln e = 1. So, 0 < 1, which holds. What if I pick a = 2 and b = 3, both less than e (~2.718). Then, b ln a = 3 ln 2 ≈ 3*0.693 ≈ 2.079, and a ln b = 2 ln 3 ≈ 2*1.098 ≈ 2.197. So, 2.079 < 2.197, which holds.Now, let's try t slightly larger than e, say t = 3. Let me pick a = e and b = 3. Then, b ln a = 3 ln e = 3*1 = 3, and a ln b = e ln 3 ≈ 2.718*1.098 ≈ 2.98. So, 3 > 2.98, which violates the inequality. So, indeed, when t exceeds e, the inequality doesn't hold for some a and b.Therefore, the maximum t is e. That makes sense because beyond e, the function f(x) = (ln x)/x starts decreasing, so the inequality flips.I think I've got it. The key was recognizing that the inequality can be transformed into a statement about the function f(x) = (ln x)/x being increasing, and then finding where that function is increasing by taking its derivative. Once I saw that the derivative is positive only when x < e, it became clear that t can't be larger than e. Testing with specific numbers helped confirm that intuition. So, yeah, the maximum value of t is e.**Final Answer**The maximum value of ( t ) is (boxed{e}).