💡First, I need to determine the total area of the walls in one bedroom. Each bedroom has four walls: two walls that are 15 feet long and 9 feet high, and two walls that are 12 feet wide and 9 feet high.Calculating the area of the two longer walls:2 × (15 feet × 9 feet) = 2 × 135 square feet = 270 square feetCalculating the area of the two shorter walls:2 × (12 feet × 9 feet) = 2 × 108 square feet = 216 square feetAdding these together gives the total wall area in one bedroom:270 square feet + 216 square feet = 486 square feetNext, I need to subtract the area occupied by doorways and windows, which is 80 square feet per bedroom:486 square feet - 80 square feet = 406 square feetSince there are four bedrooms, I multiply the paintable area of one bedroom by four:406 square feet × 4 = 1624 square feetTherefore, the total square footage of walls that must be painted is 1624.

💡Okay, so I have this problem where I need to find the minimum value of the sum of the reciprocals of eight positive real numbers, given that their sum is 2. Let me try to break this down step by step.First, let me restate the problem to make sure I understand it correctly. I have eight positive real numbers: (b_1, b_2, dots, b_8). Their sum is given as 2, so (b_1 + b_2 + dots + b_8 = 2). I need to find the minimum value of the expression (frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8}).Hmm, this seems like an optimization problem with a constraint. The constraint is that the sum of the (b_i)s is 2, and I need to minimize the sum of their reciprocals. I remember that for optimization problems with constraints, methods like Lagrange multipliers can be used, but I'm not sure if that's the easiest way here. Maybe there's a simpler inequality that applies.Wait, the Cauchy-Schwarz inequality might be useful here. I recall that Cauchy-Schwarz can relate sums and sums of reciprocals. Let me try to recall the inequality. In its general form, it states that for any real numbers (a_i) and (c_i), the following holds:[left( sum_{i=1}^{n} a_i c_i right)^2 leq left( sum_{i=1}^{n} a_i^2 right) left( sum_{i=1}^{n} c_i^2 right)]But I'm not sure if that's directly applicable here. Maybe there's another form of the inequality that's more suitable. Oh, right! There's a version of the Cauchy-Schwarz inequality that relates the sum of numbers and the sum of their reciprocals. It goes something like this:[left( sum_{i=1}^{n} b_i right) left( sum_{i=1}^{n} frac{1}{b_i} right) geq n^2]Where (n) is the number of terms. Let me verify this. If I set (a_i = sqrt{b_i}) and (c_i = frac{1}{sqrt{b_i}}), then the Cauchy-Schwarz inequality becomes:[left( sum_{i=1}^{n} a_i c_i right)^2 leq left( sum_{i=1}^{n} a_i^2 right) left( sum_{i=1}^{n} c_i^2 right)]Substituting (a_i) and (c_i), we get:[left( sum_{i=1}^{n} sqrt{b_i} cdot frac{1}{sqrt{b_i}} right)^2 leq left( sum_{i=1}^{n} b_i right) left( sum_{i=1}^{n} frac{1}{b_i} right)]Simplifying the left side:[left( sum_{i=1}^{n} 1 right)^2 = n^2]So, putting it all together:[n^2 leq left( sum_{i=1}^{n} b_i right) left( sum_{i=1}^{n} frac{1}{b_i} right)]Which rearranges to:[left( sum_{i=1}^{n} b_i right) left( sum_{i=1}^{n} frac{1}{b_i} right) geq n^2]Yes, that seems correct. So in this problem, (n = 8), and the sum of the (b_i)s is 2. Let me plug those values into the inequality:[2 times left( frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8} right) geq 8^2 = 64]So, dividing both sides by 2:[frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8} geq frac{64}{2} = 32]Okay, so this tells me that the sum of the reciprocals is at least 32. But is this the minimum value? I need to check if this bound is achievable, meaning if there exist values of (b_i) that satisfy the equality condition of the Cauchy-Schwarz inequality.The equality condition for Cauchy-Schwarz occurs when the sequences (a_i) and (c_i) are proportional. In our case, (a_i = sqrt{b_i}) and (c_i = frac{1}{sqrt{b_i}}). For these to be proportional, there must exist a constant (k) such that:[sqrt{b_i} = k times frac{1}{sqrt{b_i}} quad text{for all } i]Multiplying both sides by (sqrt{b_i}):[b_i = k]So, all (b_i) must be equal. Let me denote each (b_i = b). Since there are 8 terms, their sum is (8b = 2), so (b = frac{2}{8} = frac{1}{4}).Let me check if this gives the sum of reciprocals as 32. Each reciprocal is (frac{1}{b} = 4), and there are 8 terms, so the total sum is (8 times 4 = 32). Perfect, that matches the lower bound we found earlier.Therefore, the minimum value of the sum of reciprocals is indeed 32, achieved when all (b_i) are equal to (frac{1}{4}).Wait, just to make sure I haven't missed anything, let me consider if there's another approach, maybe using the AM-HM inequality. The Arithmetic Mean - Harmonic Mean inequality states that for positive real numbers,[frac{b_1 + b_2 + dots + b_n}{n} geq frac{n}{frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_n}}]Which can be rearranged to:[frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_n} geq frac{n^2}{b_1 + b_2 + dots + b_n}]In this case, (n = 8) and the sum of (b_i) is 2, so:[frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8} geq frac{8^2}{2} = frac{64}{2} = 32]So, this confirms the same result using a different inequality. The AM-HM inequality also tells us that equality holds when all the (b_i) are equal, which again gives each (b_i = frac{1}{4}).Just to be thorough, let me consider if there's any other method, perhaps using calculus. If I set up the problem using Lagrange multipliers, I can define the function to minimize as:[f(b_1, b_2, dots, b_8) = frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8}]Subject to the constraint:[g(b_1, b_2, dots, b_8) = b_1 + b_2 + dots + b_8 - 2 = 0]The method of Lagrange multipliers tells us that at the minimum, the gradient of (f) is proportional to the gradient of (g). So,[nabla f = lambda nabla g]Calculating the partial derivatives, for each (i):[frac{partial f}{partial b_i} = -frac{1}{b_i^2}][frac{partial g}{partial b_i} = 1]So, setting them proportional:[-frac{1}{b_i^2} = lambda times 1]Which implies that for each (i), (-frac{1}{b_i^2} = lambda). Therefore, all (b_i) must satisfy the same equation, meaning all (b_i) are equal. Let (b_i = b) for all (i), then as before, (8b = 2) so (b = frac{1}{4}).Thus, using Lagrange multipliers also leads us to the conclusion that each (b_i) must be (frac{1}{4}), and the sum of reciprocals is 32.I think I've covered multiple methods: Cauchy-Schwarz, AM-HM inequality, and Lagrange multipliers. All of them lead to the same conclusion that the minimum value is 32 when all (b_i) are equal to (frac{1}{4}).Just to ensure there are no mistakes, let me consider if making some (b_i) larger and others smaller could result in a smaller sum of reciprocals. For example, suppose I set one (b_i) to be very small, say approaching 0, then its reciprocal would approach infinity, which would make the sum of reciprocals very large. On the other hand, if I make one (b_i) very large, then its reciprocal would approach 0, but since the sum of all (b_i) is fixed at 2, making one (b_i) larger would require others to be smaller, which would increase their reciprocals. It seems that any deviation from equal values would cause the sum of reciprocals to increase, which supports the conclusion that the minimum occurs when all (b_i) are equal.Another way to think about it is that the function (f(x) = frac{1}{x}) is convex for (x > 0). By Jensen's inequality, for a convex function, the average of the function at some points is greater than or equal to the function at the average of the points. So,[frac{f(b_1) + f(b_2) + dots + f(b_8)}{8} geq fleft( frac{b_1 + b_2 + dots + b_8}{8} right)]Multiplying both sides by 8:[frac{1}{b_1} + frac{1}{b_2} + dots + frac{1}{b_8} geq 8 times fleft( frac{2}{8} right) = 8 times frac{1}{frac{1}{4}} = 8 times 4 = 32]Again, this confirms the same result. So, using Jensen's inequality also leads us to the conclusion that the minimum sum of reciprocals is 32 when all (b_i) are equal.I think I've thoroughly explored this problem from multiple angles and each approach consistently leads to the conclusion that the minimum value is 32. I don't see any flaws in the reasoning, and all the inequalities used are valid for positive real numbers, which is given in the problem. Therefore, I'm confident that the minimum value is indeed 32.**Final Answer**The minimum value is boxed{32}.

💡First, I need to determine the prime factors of each number in the product (97 cdot 101 cdot 104 cdot 107 cdot 109).Starting with 97, I recognize that 97 is a prime number because it has no divisors other than 1 and itself.Next, 101 is also a prime number since it doesn't have any divisors other than 1 and itself.Moving on to 104, I can break it down into its prime factors. Dividing 104 by 2 gives 52, and dividing 52 by 2 again gives 26. Dividing 26 by 2 once more results in 13, which is a prime number. Therefore, the prime factors of 104 are (2^3 cdot 13).Then, 107 is a prime number as it has no divisors other than 1 and itself.Lastly, 109 is also a prime number since it doesn't have any divisors other than 1 and itself.After identifying all the prime factors, I compile the list of unique primes: 97, 101, 107, 109, 2, and 13. Counting these, I find there are six distinct prime factors in total.

💡First, I need to identify the largest two-digit number that can be formed using the digits 0, 3, 5, 7, and 8 without repeating any digit. To do this, I'll choose the largest available digit for the tens place and the next largest for the ones place. The largest digit is 8, and the next largest is 7. Therefore, the largest two-digit number is 87.Next, I'll determine the smallest two-digit number using the same set of digits. For the smallest number, I should select the smallest non-zero digit for the tens place and the smallest remaining digit for the ones place. The smallest non-zero digit is 3, and the smallest remaining digit is 0. Thus, the smallest two-digit number is 30.Finally, I'll calculate the sum of these two numbers:87 (largest two-digit number) + 30 (smallest two-digit number) = 117So, the sum of the largest and smallest two-digit numbers that can be formed is 117.

💡Okay, so I have this problem about power functions and quadrants. Let me try to understand it step by step. The proposition given is: "If the function ( y = f(x) ) is a power function, then the graph of ( y = f(x) ) does not pass through the fourth quadrant." I need to figure out how many of its converse, inverse, and contrapositive propositions are true. The options are A: 0, B: 1, C: 2, D: 3.First, I should recall what a power function is. From what I remember, a power function is of the form ( f(x) = kx^n ), where ( k ) is a constant and ( n ) is an exponent. So, examples would be ( f(x) = x ), ( f(x) = x^2 ), ( f(x) = 1/x ), etc.Now, the original proposition says that if a function is a power function, then its graph doesn't go through the fourth quadrant. Let me visualize the quadrants. The fourth quadrant is where ( x ) is positive and ( y ) is negative. So, does a power function ever pass through the fourth quadrant?Let me think about some examples. Take ( f(x) = x ). That's a straight line through the origin, going through the first and third quadrants. Doesn't go through the fourth. How about ( f(x) = x^2 )? That's a parabola opening upwards, so it's in the first and second quadrants. Doesn't go through the fourth either.What about ( f(x) = 1/x )? That's a hyperbola. In the first quadrant, when ( x ) is positive, ( y ) is positive. In the third quadrant, when ( x ) is negative, ( y ) is negative. So, it doesn't pass through the fourth quadrant either. Hmm, interesting.Wait, is there any power function that does pass through the fourth quadrant? Let me think. If the exponent is negative, like ( f(x) = x^{-1} = 1/x ), as I just considered, it doesn't pass through the fourth quadrant. What if the exponent is a fraction, like ( f(x) = x^{1/2} )? That's the square root function, which is only defined for ( x geq 0 ) and ( y geq 0 ), so it doesn't go into the fourth quadrant either.What about negative exponents? For example, ( f(x) = x^{-2} = 1/x^2 ). That's always positive, so again, it doesn't go into the fourth quadrant. So, it seems like all power functions either stay in the first and second quadrants, or the first and third, or just the first quadrant, but none go into the fourth. So, the original proposition is true.Now, I need to consider the converse, inverse, and contrapositive of this proposition. Let me recall the definitions:- **Converse**: Switches the hypothesis and conclusion. So, it would be: "If the graph of ( y = f(x) ) does not pass through the fourth quadrant, then ( y = f(x) ) is a power function." - **Inverse**: Negates both the hypothesis and conclusion. So, it would be: "If the function ( y = f(x) ) is not a power function, then the graph of ( y = f(x) ) passes through the fourth quadrant." - **Contrapositive**: Negates both the hypothesis and conclusion and switches them. So, it would be: "If the graph of ( y = f(x) ) passes through the fourth quadrant, then ( y = f(x) ) is not a power function."I know that the contrapositive of a true statement is always true. So, since the original proposition is true, its contrapositive must also be true. That gives me at least one true proposition among the three.Now, let's check the converse. The converse says that if a function doesn't pass through the fourth quadrant, then it must be a power function. Hmm, is that true? Let me think of a function that doesn't pass through the fourth quadrant but isn't a power function.For example, consider the constant function ( f(x) = 2 ). Its graph is a horizontal line at ( y = 2 ), which doesn't pass through the fourth quadrant because ( y ) is always positive. But ( f(x) = 2 ) is not a power function because it's a constant function, not of the form ( kx^n ). So, the converse is false.Next, the inverse. The inverse says that if a function is not a power function, then its graph passes through the fourth quadrant. Is that true? Let's see. If I take a function that isn't a power function, does it necessarily pass through the fourth quadrant?Again, consider ( f(x) = 2 ). It's not a power function, but its graph doesn't pass through the fourth quadrant. So, this serves as a counterexample. Therefore, the inverse is also false.So, summarizing:- Original proposition: True- Converse: False- Inverse: False- Contrapositive: TrueBut the question is only asking about the converse, inverse, and contrapositive. So, among these three, only the contrapositive is true. Therefore, the number of true propositions is 1.Wait, but let me double-check. Is there any case where a non-power function doesn't pass through the fourth quadrant? For example, ( f(x) = e^x ). That's an exponential function, not a power function, and it's always positive, so it doesn't pass through the fourth quadrant either. So, that's another counterexample for the inverse.Similarly, what about a quadratic function like ( f(x) = x^2 + 1 )? It's always positive, so it doesn't pass through the fourth quadrant, but it's not a power function. So, yeah, the inverse is definitely false.And the converse, as I thought earlier, is also false because there are functions that don't pass through the fourth quadrant without being power functions.So, only the contrapositive is true. Therefore, the answer should be B: 1.**Final Answer**boxed{B}

💡Alright, so I have this problem here: Given that ( m ) and ( n ) are both positive integers, ( 2m + 3 = 5n - 2 ), and ( 5n - 2 < 15 ), find the values of ( m ) and ( n ).Hmm, okay. Let me try to break this down step by step. First, I need to understand what's being asked. We have two equations or inequalities involving ( m ) and ( n ), and we need to find their specific values given that they are positive integers.Starting with the inequality: ( 5n - 2 < 15 ). I think I should solve this first to find possible values for ( n ). So, let's work on that.Adding 2 to both sides of the inequality: ( 5n - 2 + 2 < 15 + 2 ), which simplifies to ( 5n < 17 ). Now, to solve for ( n ), I'll divide both sides by 5: ( n < frac{17}{5} ). Calculating ( frac{17}{5} ) gives me 3.4. Since ( n ) has to be a positive integer, the possible values for ( n ) are 1, 2, or 3 because 3.4 is just a bit more than 3, and ( n ) has to be less than that.Okay, so ( n ) can be 1, 2, or 3. Now, moving on to the equation ( 2m + 3 = 5n - 2 ). I need to find ( m ) in terms of ( n ) or vice versa. Let me rearrange this equation to express ( m ) in terms of ( n ).Subtracting 3 from both sides: ( 2m = 5n - 2 - 3 ), which simplifies to ( 2m = 5n - 5 ). Then, dividing both sides by 2: ( m = frac{5n - 5}{2} ).So, ( m ) is equal to ( frac{5n - 5}{2} ). Since ( m ) must also be a positive integer, ( frac{5n - 5}{2} ) must result in an integer. That means ( 5n - 5 ) must be even because dividing an even number by 2 gives an integer.Let me check for each possible value of ( n ):1. If ( n = 1 ): ( 5(1) - 5 = 5 - 5 = 0 ). Then, ( m = frac{0}{2} = 0 ). But ( m ) has to be a positive integer, so ( m = 0 ) is not acceptable.2. If ( n = 2 ): ( 5(2) - 5 = 10 - 5 = 5 ). Then, ( m = frac{5}{2} = 2.5 ). Hmm, that's not an integer. So, ( n = 2 ) doesn't work either.3. If ( n = 3 ): ( 5(3) - 5 = 15 - 5 = 10 ). Then, ( m = frac{10}{2} = 5 ). That's a positive integer, so this works.So, the only valid solution is when ( n = 3 ) and ( m = 5 ).Wait, let me double-check to make sure I didn't miss anything. I considered all possible values of ( n ) based on the inequality, and only ( n = 3 ) gave me a valid positive integer value for ( m ). For ( n = 1 ), ( m ) was zero, which isn't positive, and for ( n = 2 ), ( m ) wasn't an integer. So, yeah, ( n = 3 ) and ( m = 5 ) must be the correct answer.Just to be thorough, I'll plug these values back into the original equation to verify:( 2m + 3 = 5n - 2 )( 2(5) + 3 = 5(3) - 2 )( 10 + 3 = 15 - 2 )( 13 = 13 )Yep, that checks out. And also, checking the inequality:( 5n - 2 < 15 )( 5(3) - 2 = 15 - 2 = 13 < 15 )Which is true. So, everything seems consistent.I think I've covered all the steps and checked my work to ensure there are no mistakes. It looks like ( m = 5 ) and ( n = 3 ) are the only values that satisfy both the equation and the inequality given the constraints that ( m ) and ( n ) are positive integers.