💡First, I need to determine the total distance of the trip from Carville to Nikpath and then to Finstown. The distance from Carville to Nikpath is 315 miles, and from Nikpath to Finstown is an additional 70 miles. So, the total distance is 315 + 70 = 385 miles.Next, I need to calculate the time it takes to travel this total distance at an average speed of 60 miles per hour. Time is calculated by dividing the distance by the speed, so the driving time is 385 miles / 60 mph = 6.4167 hours.Additionally, the driver takes a 30-minute break in Nikpath. Since 30 minutes is 0.5 hours, I need to add this to the driving time. Therefore, the total time for the trip is 6.4167 hours + 0.5 hours = 6.9167 hours.Finally, I round 6.9167 hours to the nearest hundredth, which gives me 6.92 hours.

💡To find the sum of the squares of the solutions of the quadratic equation ( x^2 - 16x + 15 = 0 ), I can use Vieta's formulas. First, I identify the coefficients of the equation:- The coefficient of ( x^2 ) is 1.- The coefficient of ( x ) is -16.- The constant term is 15.Using Vieta's formulas, the sum of the roots (( s_1 + s_2 )) is equal to the negation of the coefficient of ( x ) divided by the coefficient of ( x^2 ), which gives:[s_1 + s_2 = frac{-(-16)}{1} = 16]The product of the roots (( s_1s_2 )) is equal to the constant term divided by the coefficient of ( x^2 ), which gives:[s_1s_2 = frac{15}{1} = 15]Next, I use the identity for the sum of squares of the roots:[s_1^2 + s_2^2 = (s_1 + s_2)^2 - 2s_1s_2]Substituting the known values:[s_1^2 + s_2^2 = 16^2 - 2 times 15 = 256 - 30 = 226]Therefore, the sum of the squares of the solutions is 226.

💡Okay, so I have this problem about a function f(x) that's defined on all real numbers. It satisfies the condition f(x) + f(-x) = 0, which I remember means it's an odd function. That also tells me that f(0) must be zero because if I plug in x = 0, I get f(0) + f(0) = 0, so 2f(0) = 0, hence f(0) = 0.For part (I), I need to find the explicit formula for f(x). They've given me that when x is positive, f(x) is equal to -4^x + 8*2^x + 1. So, I know what f(x) is when x is positive, but I need to figure out what it is when x is negative and when x is zero.Since f(x) is odd, I know that f(-x) = -f(x). So, if I have f(x) for positive x, then for negative x, f(x) should be the negative of f(-x). Let me write that down.If x > 0, then f(x) = -4^x + 8*2^x + 1.If x < 0, then let me set y = -x, so y > 0. Then f(x) = f(-y) = -f(y) because it's odd. So, f(x) = -f(y) = -[ -4^y + 8*2^y + 1 ].Let me compute that: -[ -4^y + 8*2^y + 1 ] = 4^y - 8*2^y - 1.But since y = -x, I can write 4^y as 4^{-x} and 2^y as 2^{-x}. So, f(x) = 4^{-x} - 8*2^{-x} - 1 when x < 0.And when x = 0, as I figured out earlier, f(0) = 0.So, putting it all together, the explicit formula for f(x) is a piecewise function:f(x) = { 4^{-x} - 8*2^{-x} - 1, when x < 0; 0, when x = 0; -4^x + 8*2^x + 1, when x > 0 }I think that's part (I) done. Now, moving on to part (II), which asks for the maximum and minimum values of f(x) when x is in the interval [-3, -1].Since x is between -3 and -1, which is all negative numbers, I need to use the expression for f(x) when x < 0, which is 4^{-x} - 8*2^{-x} - 1.Hmm, 4^{-x} is the same as (1/4)^x, and 2^{-x} is the same as (1/2)^x. Maybe I can rewrite f(x) in terms of a substitution to make it easier to handle.Let me set t = (1/2)^x. Since x is between -3 and -1, let's see what t would be.When x = -3, t = (1/2)^{-3} = 2^3 = 8.When x = -1, t = (1/2)^{-1} = 2^1 = 2.So, as x goes from -3 to -1, t goes from 8 down to 2. So, t is in [2, 8].Now, let's express f(x) in terms of t.First, 4^{-x} is (1/4)^x = ( (1/2)^2 )^x = (1/2)^{2x} = [ (1/2)^x ]^2 = t^2.Similarly, 2^{-x} is (1/2)^x = t.So, substituting back into f(x):f(x) = t^2 - 8t - 1.So, now f(x) is expressed as a quadratic function in terms of t: f(t) = t^2 - 8t - 1, where t is in [2, 8].Quadratic functions have their extrema either at the vertex or at the endpoints of the interval. The vertex of a quadratic function f(t) = at^2 + bt + c is at t = -b/(2a). In this case, a = 1, b = -8, so t = -(-8)/(2*1) = 8/2 = 4.So, the vertex is at t = 4, which is within our interval [2, 8]. So, we need to evaluate f(t) at t = 4, t = 2, and t = 8 to find the maximum and minimum.Let's compute f(4):f(4) = (4)^2 - 8*(4) - 1 = 16 - 32 - 1 = -17.f(2) = (2)^2 - 8*(2) - 1 = 4 - 16 - 1 = -13.f(8) = (8)^2 - 8*(8) - 1 = 64 - 64 - 1 = -1.So, f(t) at t = 4 is -17, which is the minimum, and at t = 8 is -1, which is the maximum.But wait, let me make sure. Since the quadratic opens upwards (because the coefficient of t^2 is positive), the vertex at t = 4 is the minimum point, and the maximum occurs at one of the endpoints. Comparing f(2) = -13 and f(8) = -1, clearly f(8) is larger. So, the maximum is -1, and the minimum is -17.Therefore, the maximum value of f(x) on [-3, -1] is -1, and the minimum value is -17.Let me just double-check my substitution and calculations to make sure I didn't make any mistakes.Starting from f(x) = 4^{-x} - 8*2^{-x} - 1 for x < 0.Set t = 2^{-x}, which is the same as (1/2)^x.So, 4^{-x} = (2^2)^{-x} = 2^{-2x} = (2^{-x})^2 = t^2.Therefore, f(x) = t^2 - 8t - 1.Yes, that seems correct.Then, since x is between -3 and -1, t = 2^{-x} is between 2^{3} = 8 and 2^{1} = 2, so t is in [2, 8]. Correct.Quadratic function f(t) = t^2 - 8t - 1.Vertex at t = 4, which is within [2, 8]. So, the minimum is at t = 4, which is f(4) = 16 - 32 - 1 = -17.At t = 2, f(2) = 4 - 16 - 1 = -13.At t = 8, f(8) = 64 - 64 - 1 = -1.So, yes, the maximum is -1, and the minimum is -17.Therefore, I think my answers are correct.**Final Answer**Ⅰ. The explicit formula for ( f(x) ) is:[f(x) = begin{cases} left(frac{1}{4}right)^x - 8 left(frac{1}{2}right)^x - 1, & x < 0 0, & x = 0 -4^x + 8 times 2^x + 1, & x > 0end{cases}]Ⅱ. The maximum value of ( f(x) ) on ( [-3, -1] ) is (boxed{-1}) and the minimum value is (boxed{-17}).

💡Okay, so I've got this problem here: A square is divided into 5 identical smaller rectangles. Each of these smaller rectangles has a perimeter of 120 centimeters. I need to find the area of the original square. Hmm, let me try to figure this out step by step.First, I know that a square has all sides equal, and when it's divided into smaller rectangles, those rectangles must fit perfectly within the square. Since there are 5 identical rectangles, I can imagine that they might be arranged either in a single row or column, or maybe in some other configuration. But given that it's a square, the arrangement has to maintain the equal sides.Let me denote the length and width of each smaller rectangle as ( l ) and ( w ) respectively. The perimeter of a rectangle is given by the formula ( 2(l + w) ). According to the problem, each rectangle has a perimeter of 120 cm. So, I can write the equation:[ 2(l + w) = 120 ]Dividing both sides by 2 to simplify:[ l + w = 60 ]Okay, so the sum of the length and width of each small rectangle is 60 cm. Now, since the square is divided into 5 identical rectangles, I need to figure out how these rectangles are arranged. One common way to divide a square into identical rectangles is to have them arranged in a single row or column. Let me consider both possibilities.**Case 1: The rectangles are arranged in a single row.**If the rectangles are placed side by side in a single row, the length of each rectangle would be equal to the side of the square, and the width would be such that 5 widths make up the side of the square. Wait, that might not make sense. Let me think again.Actually, if they're arranged in a single row, the total length would be 5 times the width of one rectangle, and the height would be equal to the length of one rectangle. But since it's a square, the total length and height must be equal. So, if the rectangles are arranged in a single row, the length of the square would be ( 5w ) and the height would be ( l ). Therefore:[ 5w = l ]So, the length of each rectangle is 5 times its width. That gives me another equation:[ l = 5w ]Now, I can substitute this into the earlier equation ( l + w = 60 ):[ 5w + w = 60 ][ 6w = 60 ][ w = 10 ]So, the width of each rectangle is 10 cm. Then, the length ( l ) would be:[ l = 5w = 5 times 10 = 50 text{ cm} ]Now, since the square is made up of these rectangles arranged in a single row, the side of the square would be equal to the length of the rectangle, which is 50 cm. Therefore, the area of the square would be:[ text{Area} = text{side}^2 = 50^2 = 2500 text{ cm}^2 ]Wait, that seems straightforward. But let me check if there's another possible arrangement.**Case 2: The rectangles are arranged in a single column.**If the rectangles are stacked vertically, then the height of the square would be 5 times the height of one rectangle, and the width of the square would be equal to the width of one rectangle. But since it's a square, the width and height must be equal. So, if the rectangles are arranged in a single column, the height of the square would be ( 5l ) and the width would be ( w ). Therefore:[ 5l = w ]But wait, that would mean the width of the square is much larger than the height, which contradicts the fact that it's a square. So, this arrangement doesn't make sense because it would result in a rectangle, not a square. Therefore, the only feasible arrangement is the first case where the rectangles are arranged in a single row.Just to be thorough, let me consider if there's another way to divide the square into 5 rectangles. Maybe in a 2x2 grid with one extra rectangle? But that would complicate things and likely not result in identical rectangles. So, I think the initial assumption of arranging them in a single row is correct.Therefore, with the side of the square being 50 cm, the area is indeed 2500 cm².**Final Answer**The area of the original square is boxed{2500} square centimeters.

💡Okay, so I need to find the fraction that's exactly halfway between 3/4 and 5/6. Hmm, I think this means I need to find the average of these two fractions. But wait, how do I average fractions? I remember that to average two numbers, you add them together and then divide by 2. So, I guess I need to add 3/4 and 5/6 and then divide the result by 2. But before I can add them, I need to make sure they have the same denominator. The denominators here are 4 and 6. What's the least common denominator for these two? Well, 4 and 6 both go into 12, so I'll convert both fractions to have 12 as the denominator. For 3/4, if I multiply both the numerator and the denominator by 3, I get 9/12. And for 5/6, if I multiply both the numerator and the denominator by 2, I get 10/12. Okay, so now I have 9/12 and 10/12. Now, I need to add these two fractions together. Adding 9/12 and 10/12 is straightforward because they have the same denominator. I just add the numerators: 9 + 10 equals 19, so the sum is 19/12. But wait, I need to find the average, which means I have to divide this sum by 2. So, I take 19/12 and divide it by 2. Dividing by 2 is the same as multiplying by 1/2, so 19/12 times 1/2 equals 19/24. Let me double-check my steps to make sure I didn't make a mistake. I converted both fractions to have a common denominator, added them, and then divided by 2. That seems right. So, the fraction exactly halfway between 3/4 and 5/6 is 19/24.

💡Alright, so I have this problem here where I need to show that the sum of ( a_i b_i ) is less than or equal to the sum of ( a_i B_i ). Let me try to understand what's given and what I need to prove.First, I have two sequences: one is ( a_1 geq a_2 geq ldots geq a_n ), which is a sequence of real numbers in non-increasing order. Then, there's another sequence ( b_1, b_2, ldots, b_n ), which is just a rearrangement of the ( a ) sequence. So, the ( b )s are the same numbers as the ( a )s but possibly in a different order. Then, ( B_1 geq B_2 geq ldots geq B_n ) is another sequence, which is also in non-increasing order, but it's not specified whether it's related to the ( a )s or ( b )s.Wait, actually, the problem says ( b_1, b_2, ldots, b_n ) is any rearrangement of the sequence. Hmm, does that mean it's a rearrangement of the ( a ) sequence? Or is it a rearrangement of another sequence? Let me read that again.It says: " ( b_1, b_2, b_3, ldots, b_n ) is any rearrangement of the sequence. ( B_1 geq B_2 geq ldots geq B_n )." So, it seems like ( b ) is a rearrangement of the original sequence, which is the ( a ) sequence. So, ( b ) is just a permutation of ( a ). Then, ( B ) is another sequence that is sorted in non-increasing order, but it's not specified whether it's related to ( a ) or ( b ). Maybe ( B ) is the sorted version of ( b )? Or is it a different sequence altogether?Wait, the problem says " ( b_1, b_2, b_3, ldots, b_n ) is any rearrangement of the sequence. ( B_1 geq B_2 geq ldots geq B_n )." So, perhaps ( B ) is the sorted version of ( b )? That would make sense because ( B ) is in non-increasing order. So, if ( b ) is any rearrangement, then ( B ) is the sorted version of ( b ). So, ( B ) is the same as ( a ), but sorted in non-increasing order.Wait, but ( a ) is already given as ( a_1 geq a_2 geq ldots geq a_n ). So, if ( b ) is a rearrangement of ( a ), then ( B ) is just ( a ) itself because ( a ) is already sorted. But that can't be because then ( B ) would be the same as ( a ), and the sum ( sum a_i B_i ) would just be ( sum a_i^2 ), which might not necessarily be greater than ( sum a_i b_i ).Hmm, maybe I misinterpreted. Let me read the problem again carefully:" ( a_1 geq a_2 geq ldots geq a_n ) is a sequence of real numbers. ( b_1, b_2, b_3, ldots, b_n ) is any rearrangement of the sequence. ( B_1 geq B_2 geq ldots geq B_n ). Show that ( sum a_i b_i leq sum a_i B_i )."So, ( a ) is a sequence in non-increasing order. ( b ) is any rearrangement of ( a ), so ( b ) is a permutation of ( a ). Then, ( B ) is another sequence in non-increasing order, but it's not specified whether it's related to ( a ) or ( b ). Wait, maybe ( B ) is the sorted version of ( b )? Because ( b ) is a rearrangement, so ( B ) would be the sorted version of ( b ), which would be the same as ( a ), since ( a ) is already sorted.Wait, that would mean ( B ) is equal to ( a ), so ( sum a_i B_i = sum a_i^2 ). But then, ( sum a_i b_i ) is the sum of ( a_i ) times a permutation of ( a ). So, the problem is to show that the sum of ( a_i b_i ) is less than or equal to the sum of ( a_i^2 ).But that doesn't sound right because if ( b ) is a permutation of ( a ), then ( sum a_i b_i ) can be equal to ( sum a_i^2 ) if ( b ) is the same as ( a ), but if ( b ) is a different permutation, it might be less. So, maybe the problem is to show that the sum is maximized when ( b ) is sorted in the same order as ( a ).Wait, that makes sense. So, if you have two sequences, both sorted in non-increasing order, their sum of products is maximized when they are similarly ordered. This is known as the rearrangement inequality. So, if ( a ) is sorted in non-increasing order and ( B ) is also sorted in non-increasing order, then ( sum a_i B_i ) is the maximum possible sum when both sequences are similarly ordered. Any other rearrangement ( b ) of ( a ) would result in a sum ( sum a_i b_i ) that is less than or equal to ( sum a_i B_i ).So, the key here is to apply the rearrangement inequality, which states that for two sequences sorted in the same order, their sum of products is maximized when they are similarly ordered, and minimized when they are opposely ordered.Therefore, since ( B ) is the sorted version of ( b ), which is a rearrangement of ( a ), and ( a ) is already sorted, the sum ( sum a_i B_i ) is the maximum possible sum, and any other rearrangement ( b ) would give a sum less than or equal to that.But let me make sure I'm not missing anything. Let me think through an example.Suppose ( a = [3, 2, 1] ), so ( a_1 = 3, a_2 = 2, a_3 = 1 ). Then, a rearrangement ( b ) could be [2, 3, 1]. So, ( b_1 = 2, b_2 = 3, b_3 = 1 ). Then, ( B ) would be the sorted version of ( b ), which is [3, 2, 1], same as ( a ). So, ( sum a_i b_i = 3*2 + 2*3 + 1*1 = 6 + 6 + 1 = 13 ). ( sum a_i B_i = 3*3 + 2*2 + 1*1 = 9 + 4 + 1 = 14 ). So, indeed, 13 ≤ 14.Another example: ( a = [5, 4, 3, 2, 1] ). Let ( b = [2, 3, 5, 1, 4] ). Then, ( B ) would be [5, 4, 3, 2, 1]. So, ( sum a_i b_i = 5*2 + 4*3 + 3*5 + 2*1 + 1*4 = 10 + 12 + 15 + 2 + 4 = 43 ). ( sum a_i B_i = 5*5 + 4*4 + 3*3 + 2*2 + 1*1 = 25 + 16 + 9 + 4 + 1 = 55 ). So, 43 ≤ 55.This seems to hold. So, the idea is that when both sequences are sorted in the same order, their product sum is maximized. Any permutation of one sequence will result in a smaller or equal sum.Therefore, the proof should involve applying the rearrangement inequality, which is a well-known result in mathematics. The rearrangement inequality states that for two sequences of real numbers, the sum of their products is maximized when both sequences are similarly ordered and minimized when they are opposely ordered.In this case, since ( a ) is already sorted in non-increasing order, and ( B ) is the sorted version of ( b ), which is a rearrangement of ( a ), then ( B ) is also sorted in non-increasing order. Therefore, ( sum a_i B_i ) is the maximum possible sum, and any other rearrangement ( b ) will give a sum ( sum a_i b_i ) that is less than or equal to ( sum a_i B_i ).I think that's the gist of it. To make this rigorous, I should probably state the rearrangement inequality formally and then apply it to the given sequences.So, let me recall the rearrangement inequality:Given two sequences of real numbers ( x_1 geq x_2 geq ldots geq x_n ) and ( y_1 geq y_2 geq ldots geq y_n ), the sum ( sum_{i=1}^n x_i y_i ) is maximized when both sequences are similarly ordered and minimized when they are opposely ordered.In our case, ( a ) is already sorted in non-increasing order, and ( B ) is the sorted version of ( b ), which is a rearrangement of ( a ). Therefore, ( B ) is also sorted in non-increasing order. Thus, ( sum a_i B_i ) is the maximum possible sum, and any other rearrangement ( b ) will give a sum ( sum a_i b_i ) that is less than or equal to ( sum a_i B_i ).Therefore, the inequality ( sum a_i b_i leq sum a_i B_i ) holds by the rearrangement inequality.I think that's a solid argument. I don't see any flaws in this reasoning. The key was recognizing that ( B ) is the sorted version of ( b ), which is a rearrangement of ( a ), and applying the rearrangement inequality to compare the sums.Just to be thorough, let me consider another example where ( b ) is the reverse of ( a ). Suppose ( a = [4, 3, 2, 1] ), so ( a_1 = 4, a_2 = 3, a_3 = 2, a_4 = 1 ). Let ( b ) be the reverse, so ( b = [1, 2, 3, 4] ). Then, ( B ), being the sorted version of ( b ), is [4, 3, 2, 1], same as ( a ). So, ( sum a_i b_i = 4*1 + 3*2 + 2*3 + 1*4 = 4 + 6 + 6 + 4 = 20 ). ( sum a_i B_i = 4*4 + 3*3 + 2*2 + 1*1 = 16 + 9 + 4 + 1 = 30 ). So, 20 ≤ 30, which holds.Another test case: ( a = [10, 5, 3, 1] ). Let ( b = [5, 10, 1, 3] ). Then, ( B ) is [10, 5, 3, 1]. So, ( sum a_i b_i = 10*5 + 5*10 + 3*1 + 1*3 = 50 + 50 + 3 + 3 = 106 ). ( sum a_i B_i = 10*10 + 5*5 + 3*3 + 1*1 = 100 + 25 + 9 + 1 = 135 ). So, 106 ≤ 135, which is true.These examples reinforce the idea that the sum is maximized when both sequences are similarly ordered. Therefore, the inequality holds.I think I've covered the necessary ground here. The main takeaway is recognizing the application of the rearrangement inequality and understanding the roles of the sequences ( a ), ( b ), and ( B ). By ensuring that ( B ) is the sorted version of ( b ), we can apply the inequality to establish the desired result.