💡First, convert the distance between Alice's and Zoey's houses from miles to feet. Since 1 mile equals 5280 feet, the total distance is:[3 times 5280 = 15840 text{ feet}]Let Alice's walking speed be ( a ) feet per minute. Zoey's biking speed is then ( 4a ) feet per minute. Together, their combined speed is:[a + 4a = 5a text{ feet per minute}]They start moving towards each other, and the time ( t ) it takes for them to meet is given by:[5a times t = 15840]Solving for ( t ):[t = frac{15840}{5a} = frac{3168}{a} text{ minutes}]Alice stops for 5 minutes to rest at the halfway point. The halfway distance is:[frac{15840}{2} = 7920 text{ feet}]The time Alice takes to reach the halfway point is:[frac{7920}{a} text{ minutes}]If this time is less than or equal to the total time ( t ), Alice will have already rested before meeting Zoey. Otherwise, she will still be resting when they meet. Assuming Alice's speed ( a ) is such that she reaches the halfway point before the total meeting time, the effective time Alice spends walking is:[t_{text{walk}} = t - 5 = frac{3168}{a} - 5 text{ minutes}]The distance Alice covers by the time they meet is:[a times t_{text{walk}} = a left( frac{3168}{a} - 5 right) = 3168 - 5a text{ feet}]The number of steps Alice takes is:[frac{3168 - 5a}{2}]Assuming a typical walking speed ( a = 300 ) feet per minute:[frac{3168 - 5 times 300}{2} = frac{3168 - 1500}{2} = frac{1668}{2} = 834]Thus, the number of steps Alice will take by the time she meets Zoey is:[boxed{textbf{(B)} 834}]

💡First, I need to calculate the sum inside the parentheses: 71 + 75 + 79 + 83 + 87 + 91. Adding these numbers step by step:71 + 75 = 146146 + 79 = 225225 + 83 = 308308 + 87 = 395395 + 91 = 486Now, I have to multiply this sum by 3:3 × 486 = 1458So, the value of 3(71 + 75 + 79 + 83 + 87 + 91) is 1458.

💡Okay, so I have this triangle ABC, and the angles are given as angle A is 120 degrees, angle B is 45 degrees, and angle C is 15 degrees. The question is asking me to find the value of cos B. Hmm, okay, let's see.First, I remember that in any triangle, the sum of the angles should be 180 degrees. Let me check if that's the case here. So, 120 plus 45 is 165, and then adding 15 gives exactly 180 degrees. Okay, that checks out. So, the triangle is valid.Now, the question is about finding cos B. Since angle B is 45 degrees, I think I can directly find the cosine of 45 degrees. Wait, is that right? I mean, do I need to use the Law of Cosines here or something else? Let me think.I recall that the Law of Cosines is used when we have sides and angles, but in this case, we're only dealing with angles. Since we're given angle B and we need to find its cosine, maybe I don't need to involve the sides at all. I think it's straightforward.So, cos B is just cos 45 degrees. Now, what is the value of cos 45 degrees? I remember that cos 45 is one of those standard angles, like 30, 45, 60 degrees, which have known cosine values. Let me recall the exact value.I think cos 45 degrees is equal to sqrt(2)/2. Yeah, that sounds right. Let me visualize the unit circle. At 45 degrees, which is pi/4 radians, the coordinates on the unit circle are (sqrt(2)/2, sqrt(2)/2). So, the x-coordinate is cos 45, which is sqrt(2)/2. Okay, that makes sense.Wait, but I have to make sure I'm not confusing it with sin 45. Is sin 45 also sqrt(2)/2? Yes, it is. So, both sine and cosine of 45 degrees are equal because it's a 45-45-90 triangle, which is an isosceles right triangle. So, both legs are equal, and the hypotenuse is sqrt(2) times the leg. Therefore, the ratios for sine and cosine are both sqrt(2)/2.Alright, so going back to the problem. Since angle B is 45 degrees, cos B is sqrt(2)/2. Let me check the answer choices to see if that's one of them.Looking at the options: A is sqrt(3)/2, B is 1/2, C is sqrt(3), and D is sqrt(2)/2. So, D is the one that matches sqrt(2)/2. Therefore, the answer should be D.But wait, let me make sure I didn't miss anything. Is there a possibility that I need to use the Law of Cosines here? For example, if I had sides involved, I might need to relate them. But in this case, the problem is only about angle B, and it's given directly as 45 degrees. So, I think it's safe to go with the straightforward approach.Just to double-check, if I were to use the Law of Cosines for angle B, it would be:cos B = (a² + c² - b²) / (2ac)But wait, I don't have the sides a, b, c. All I have are the angles. So, without knowing the sides, I can't apply the Law of Cosines here. Therefore, my initial thought was correct; I don't need to use the Law of Cosines because I don't have the necessary side lengths.Alternatively, could I use the Law of Sines? The Law of Sines relates the sides and angles of a triangle, stating that a/sin A = b/sin B = c/sin C. But again, without knowing at least one side, I can't find the others. So, that's not helpful here.Therefore, I'm confident that cos B is simply the cosine of 45 degrees, which is sqrt(2)/2, corresponding to option D.Just to recap, the steps are:1. Confirm that the sum of angles in the triangle is 180 degrees.2. Recognize that angle B is 45 degrees.3. Recall the value of cos 45 degrees, which is sqrt(2)/2.4. Match this value with the given answer choices.5. Conclude that the correct answer is D.I think that's thorough enough. I don't see any mistakes in my reasoning, and all the steps make sense. So, yeah, I'm pretty sure D is the right answer.

💡Alright, so I have this problem about the function ( f(x) = frac{ln x}{x} ). It has two parts: first, I need to arrange ( f(2) ), ( f(3) ), and ( f(5) ) in ascending order, and second, I need to find the maximum value of the function ( g(x) = x^{frac{1}{x}} ) for ( x > 0 ). Hmm, okay, let's take this step by step.Starting with the first part: arranging ( f(2) ), ( f(3) ), and ( f(5) ) in ascending order. I remember that to compare function values, sometimes it's helpful to look at the behavior of the function—like whether it's increasing or decreasing. Maybe I should find the derivative of ( f(x) ) to understand its behavior better.So, ( f(x) = frac{ln x}{x} ). To find the derivative, I can use the quotient rule. The quotient rule says that if I have a function ( frac{u}{v} ), its derivative is ( frac{u'v - uv'}{v^2} ). Applying that here, where ( u = ln x ) and ( v = x ), I get:( f'(x) = frac{(1/x) cdot x - ln x cdot 1}{x^2} )Simplifying that:( f'(x) = frac{1 - ln x}{x^2} )Okay, so the derivative is ( frac{1 - ln x}{x^2} ). Now, to find where the function is increasing or decreasing, I need to look at the sign of the derivative. The denominator ( x^2 ) is always positive for ( x > 0 ), so the sign of ( f'(x) ) depends on the numerator ( 1 - ln x ).Setting the numerator equal to zero to find critical points:( 1 - ln x = 0 )( ln x = 1 )( x = e ) (since ( e ) is approximately 2.718)So, the critical point is at ( x = e ). Now, let's analyze the intervals around ( e ):1. For ( x < e ), say ( x = 2 ), ( ln 2 ) is about 0.693, which is less than 1. So, ( 1 - ln x ) is positive. Therefore, ( f'(x) > 0 ) when ( x < e ), meaning the function is increasing on ( (0, e) ).2. For ( x > e ), say ( x = 3 ), ( ln 3 ) is about 1.098, which is greater than 1. So, ( 1 - ln x ) is negative. Therefore, ( f'(x) < 0 ) when ( x > e ), meaning the function is decreasing on ( (e, infty) ).So, ( f(x) ) increases up to ( x = e ) and then decreases after that. That means the maximum value of ( f(x) ) occurs at ( x = e ).Now, looking at the points we need to compare: 2, 3, and 5. Since ( e ) is approximately 2.718, 2 is less than ( e ), 3 is greater than ( e ), and 5 is also greater than ( e ). Since the function is increasing up to ( e ), ( f(2) < f(e) ). Then, after ( e ), the function starts decreasing, so ( f(3) < f(e) ) and ( f(5) < f(e) ). But we need to compare ( f(2) ), ( f(3) ), and ( f(5) ) among themselves.Let's compute the actual values to get a clearer picture.First, ( f(2) = frac{ln 2}{2} approx frac{0.693}{2} approx 0.3465 ).Next, ( f(3) = frac{ln 3}{3} approx frac{1.0986}{3} approx 0.3662 ).Then, ( f(5) = frac{ln 5}{5} approx frac{1.6094}{5} approx 0.3219 ).So, comparing these approximate values:( f(5) approx 0.3219 ),( f(2) approx 0.3465 ),( f(3) approx 0.3662 ).Therefore, in ascending order: ( f(5) < f(2) < f(3) ).Wait, but let me double-check these calculations because sometimes approximations can be misleading. Maybe I should compute the exact differences or use logarithmic identities to compare them without approximating.Let's see:Compare ( f(2) ) and ( f(5) ):( f(2) = frac{ln 2}{2} ),( f(5) = frac{ln 5}{5} ).To compare these, let's find ( f(2) - f(5) ):( frac{ln 2}{2} - frac{ln 5}{5} = frac{5 ln 2 - 2 ln 5}{10} ).Compute the numerator:( 5 ln 2 - 2 ln 5 = ln(2^5) - ln(5^2) = ln(32) - ln(25) = lnleft(frac{32}{25}right) ).Since ( frac{32}{25} = 1.28 ), which is greater than 1, ( ln(1.28) ) is positive. Therefore, ( f(2) - f(5) > 0 ), so ( f(2) > f(5) ).Now, compare ( f(2) ) and ( f(3) ):( f(2) = frac{ln 2}{2} ),( f(3) = frac{ln 3}{3} ).Compute ( f(2) - f(3) ):( frac{ln 2}{2} - frac{ln 3}{3} = frac{3 ln 2 - 2 ln 3}{6} ).Numerator:( 3 ln 2 - 2 ln 3 = ln(2^3) - ln(3^2) = ln(8) - ln(9) = lnleft(frac{8}{9}right) ).Since ( frac{8}{9} ) is less than 1, ( lnleft(frac{8}{9}right) ) is negative. Therefore, ( f(2) - f(3) < 0 ), so ( f(2) < f(3) ).Putting it all together:( f(5) < f(2) < f(3) ).Okay, that seems consistent with my earlier approximation.Now, moving on to the second part: finding the maximum value of ( g(x) = x^{frac{1}{x}} ) for ( x > 0 ).Hmm, ( g(x) = x^{1/x} ). To find its maximum, I might need to use calculus again. Since it's an exponential function with both the base and the exponent involving ( x ), it might be tricky to differentiate directly. Maybe taking the natural logarithm would simplify things.Let me set ( y = g(x) = x^{1/x} ). Taking the natural logarithm of both sides:( ln y = frac{ln x}{x} ).Wait a minute, that's exactly the function ( f(x) ) from the first part! So, ( ln y = f(x) ).Therefore, maximizing ( y ) is equivalent to maximizing ( f(x) ), since the natural logarithm is a monotonically increasing function. So, the maximum of ( g(x) ) occurs at the same ( x ) where ( f(x) ) is maximized.From the first part, we found that ( f(x) ) attains its maximum at ( x = e ). Therefore, ( g(x) ) also attains its maximum at ( x = e ).So, the maximum value of ( g(x) ) is ( g(e) = e^{1/e} ).Let me verify this. If ( x = e ), then ( g(e) = e^{1/e} ). Is this indeed the maximum?Yes, because as ( x ) approaches 0 from the right, ( x^{1/x} ) approaches 0 (since ( 1/x ) becomes very large, and any number less than 1 raised to a large power approaches 0). As ( x ) approaches infinity, ( x^{1/x} ) approaches 1 (since ( 1/x ) approaches 0, and any number raised to the power of 0 is 1). Therefore, the function ( g(x) ) starts at 0, increases to a maximum at ( x = e ), and then decreases towards 1 as ( x ) increases further.Therefore, the maximum value is indeed ( e^{1/e} ).So, summarizing:1. The ascending order of ( f(2) ), ( f(3) ), and ( f(5) ) is ( f(5) < f(2) < f(3) ).2. The maximum value of ( g(x) = x^{1/x} ) is ( e^{1/e} ).**Final Answer**1. The ascending order is boxed{f(5) < f(2) < f(3)}.2. The maximum value of ( g(x) ) is boxed{e^{frac{1}{e}}}.

💡Okay, so I need to figure out how many stars Tilly counts in total. Let me read the problem again to make sure I understand it correctly. It says, "Tilly counts 120 stars to the east of her house and six times that number to the west. How many stars does she count total?"Alright, so Tilly is counting stars in two directions: east and west of her house. She counts 120 stars to the east. Then, to the west, she counts six times that number. I need to find the total number of stars she counts altogether.First, let's break down the information given. To the east, it's straightforward: 120 stars. To the west, it's six times the number she counted to the east. So, I need to calculate six times 120 to find out how many stars she saw to the west.Let me write that out mathematically. If the number of stars to the east is 120, then the number to the west is 6 multiplied by 120. So, that would be:Number of stars to the west = 6 × 120Now, I need to compute that. 6 times 120. Hmm, let's see. 6 times 100 is 600, and 6 times 20 is 120. So, adding those together, 600 + 120 equals 720. So, Tilly counts 720 stars to the west.Now, to find the total number of stars she counts, I need to add the stars to the east and the stars to the west together. That would be:Total stars = Stars to the east + Stars to the westTotal stars = 120 + 720Let me add those numbers. 120 plus 720. Well, 120 plus 700 is 820, and then adding the remaining 20 gives me 840. So, the total number of stars Tilly counts is 840.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the number of stars to the west: 6 times 120. 6 times 100 is 600, and 6 times 20 is 120. Adding those gives 720, which seems correct. Then, adding 120 to 720: 120 plus 700 is 820, plus 20 is 840. Yep, that checks out.I think I've got it right. Tilly counts a total of 840 stars when she looks both east and west of her house.

💡Alright, so I've got this problem here about a ship and its cargo. Let me try to understand what's going on. The ship starts off with 5973 tons of cargo. Okay, that's a pretty big number, but I guess ships can carry a lot of stuff. Then, it stops in the Bahamas, and sailors load an additional 8723 tons of cargo onto the ship. The question is asking how much cargo the ship holds now. So, basically, I need to figure out the total cargo after adding these two amounts together.First, I'll write down the numbers to make it clearer:- Initial cargo: 5973 tons- Cargo loaded in the Bahamas: 8723 tonsOkay, so to find the total cargo now, I need to add these two numbers together. That seems straightforward, but let me make sure I'm doing it right. Adding large numbers can sometimes be tricky, especially if there are any carries or borrows involved. But in this case, it's just a simple addition.So, let's set it up:``` 5973+ 8723-------```Starting from the rightmost digit, which is the units place:3 (from 5973) + 3 (from 8723) = 6. So, I write down 6.Next, the tens place:7 (from 5973) + 2 (from 8723) = 9. So, I write down 9.Moving to the hundreds place:9 (from 5973) + 7 (from 8723) = 16. Okay, that's more than 10, so I need to carry over the extra. I'll write down 6 and carry over 1 to the thousands place.Finally, the thousands place:5 (from 5973) + 8 (from 8723) = 13, plus the carried over 1 makes it 14. So, I write down 14.Putting it all together, the total is 14696 tons.Wait a second, let me double-check that to make sure I didn't make any mistakes. Maybe I should add them again to confirm.Starting over:5973 + 8723.Adding the units place: 3 + 3 = 6.Tens place: 7 + 2 = 9.Hundreds place: 9 + 7 = 16. So, 6 with a carryover of 1.Thousands place: 5 + 8 = 13, plus the carryover makes 14.Yes, that still gives me 14696 tons. Okay, that seems consistent.Just to be extra sure, maybe I can break it down differently. Let's see:5973 can be broken down into 5000 + 900 + 70 + 3.8723 can be broken down into 8000 + 700 + 20 + 3.Now, adding the thousands: 5000 + 8000 = 13000.Adding the hundreds: 900 + 700 = 1600.Adding the tens: 70 + 20 = 90.Adding the units: 3 + 3 = 6.Now, adding all these together:13000 + 1600 = 14600.14600 + 90 = 14690.14690 + 6 = 14696.Yep, that still checks out. So, the total cargo the ship holds now is 14696 tons.I think that's pretty much it. I've added the numbers in two different ways to make sure I didn't make any mistakes, and both methods gave me the same result. So, I'm confident that the answer is correct.