💡First, I need to determine how many bite-size samples can be obtained from one apple pie. Each apple pie can be divided into 2 halves, and each half can be split into 5 bite-size samples. Therefore, one apple pie can provide a total of 2 multiplied by 5, which equals 10 bite-size samples.Next, I know that Sedrach wants to let 130 people taste his apple pies by providing each person with one bite-size sample. To find out how many apple pies are needed to serve 130 people, I divide the total number of samples required by the number of samples that can be obtained from one apple pie.So, 130 samples divided by 10 samples per apple pie equals 13 apple pies. Therefore, Sedrach has 13 apple pies.

💡Alright, so I have this problem here where I need to figure out the relationship between three numbers: (a = 2^{0.3}), (b = 0.3^{2}), and (c = log_{2}0.3). The options are A, B, C, D, each giving a different order from smallest to largest. Let me try to break this down step by step.First, let's understand each of these expressions individually. Starting with (a = 2^{0.3}). I know that any number raised to the power of 0 is 1, and as the exponent increases, the result increases as well. Since 0.3 is greater than 0, (2^{0.3}) should be greater than 1. To get a better sense, maybe I can approximate it. I remember that (2^{0.3}) is the same as the 10th root of (2^3), but that might be too complicated. Alternatively, I can think of (2^{0.3}) as approximately (1 + 0.3 times ln 2), using the Taylor series expansion for small exponents. Calculating that, (ln 2) is roughly 0.693, so (0.3 times 0.693) is about 0.2079. Adding that to 1 gives approximately 1.2079. So, (a) is roughly 1.2079.Next, (b = 0.3^{2}). That's straightforward, squaring 0.3. So, 0.3 times 0.3 is 0.09. So, (b = 0.09).Now, (c = log_{2}0.3). This is a logarithm with base 2 of a number less than 1. I remember that logarithms of numbers less than 1 are negative. To get a better idea, maybe I can convert it to natural logarithm or common logarithm. Using the change of base formula, (log_{2}0.3 = frac{ln 0.3}{ln 2}). Calculating (ln 0.3), I know that (ln 1 = 0), and (ln 0.3) is negative. Specifically, (ln 0.3) is approximately -1.2039. Dividing that by (ln 2), which is about 0.693, gives roughly -1.2039 / 0.693 ≈ -1.737. So, (c) is approximately -1.737.Now, let's summarize the approximate values:- (a ≈ 1.2079)- (b = 0.09)- (c ≈ -1.737)So, arranging them from smallest to largest: (c < b < a). That would be option B.Wait, let me double-check my calculations to make sure I didn't make any mistakes. For (a = 2^{0.3}), I used the approximation (1 + 0.3 times ln 2). Is that a good approximation? Actually, the Taylor series for (e^x) is (1 + x + x^2/2 + ...), but (2^{0.3}) is (e^{0.3 ln 2}), so maybe I should use a better approximation. Let's compute (0.3 ln 2) which is approximately 0.3 * 0.693 ≈ 0.2079. Then, (e^{0.2079}) is approximately 1.230. So, my initial approximation was a bit low, but it's still around 1.23. So, (a) is approximately 1.23.For (b = 0.3^2), that's definitely 0.09.For (c = log_{2}0.3), I got approximately -1.737. Let me verify that with another method. Since (2^{-1} = 0.5) and (2^{-2} = 0.25). Since 0.3 is between 0.25 and 0.5, the logarithm should be between -2 and -1. Specifically, since 0.3 is closer to 0.25, it should be closer to -2. So, my calculation of approximately -1.737 seems reasonable.So, putting it all together, (c ≈ -1.737), (b = 0.09), and (a ≈ 1.23). Therefore, the order is (c < b < a), which is option B.I think that's solid. I don't see any mistakes in my reasoning. The key was recognizing that (a) is greater than 1, (b) is between 0 and 1, and (c) is negative. So, their order from smallest to largest is (c < b < a).**Final Answer**boxed{B}

💡Okay, so I have this problem about Lipschitz conditions. I remember that a function satisfies the Lipschitz condition on a domain if there's a constant ( k ) such that the absolute difference in function values is bounded by ( k ) times the absolute difference in inputs. Part (1) asks for an example of a function that satisfies the Lipschitz condition and to state the value of ( k ), then verify it. Hmm, Lipschitz functions are functions that don't change too quickly. So, maybe a linear function? Like ( f(x) = mx + b ). For linear functions, the slope is constant, so the Lipschitz constant ( k ) would just be the absolute value of the slope. Let me pick ( f(x) = x ), so ( m = 1 ). Then, ( |f(x_1) - f(x_2)| = |x_1 - x_2| ), which is clearly ( 1 times |x_1 - x_2| ). So, ( k = 1 ) works here. That seems straightforward.Now, part (2) is about the function ( f(x) = sqrt{x + 1} ) on the domain ([0, +infty)). I need to find the minimum value of ( k ) such that the Lipschitz condition holds. I think I can use the Mean Value Theorem here. The theorem says that for a function that's continuous on ([a, b]) and differentiable on ((a, b)), there's some ( c ) in ((a, b)) such that ( f(b) - f(a) = f'(c)(b - a) ). So, applying this to ( f(x) = sqrt{x + 1} ), the derivative ( f'(x) = frac{1}{2sqrt{x + 1}} ). To find the Lipschitz constant ( k ), I need the maximum value of ( |f'(x)| ) on the domain ([0, +infty)). Since ( f'(x) ) is positive and decreasing as ( x ) increases, its maximum occurs at the smallest ( x ), which is ( x = 0 ). Plugging in, ( f'(0) = frac{1}{2sqrt{0 + 1}} = frac{1}{2} ). So, the maximum slope is ( frac{1}{2} ), which means ( k = frac{1}{2} ) is the smallest constant that satisfies the Lipschitz condition for this function. Wait, let me double-check. If I take two points, say ( x_1 = 0 ) and ( x_2 = 1 ), then ( |f(x_1) - f(x_2)| = |sqrt{1} - sqrt{2}| = sqrt{2} - 1 approx 0.414 ). The difference ( |x_1 - x_2| = 1 ). So, ( 0.414 leq k times 1 ), which suggests ( k ) needs to be at least 0.414. But since ( frac{1}{2} = 0.5 ) is larger, it still works. If I take points closer to each other, the ratio ( frac{|f(x_1) - f(x_2)|}{|x_1 - x_2|} ) approaches ( f'(x) ), which is ( frac{1}{2sqrt{x + 1}} ). As ( x ) increases, this ratio decreases, so the maximum ratio is indeed at ( x = 0 ). Therefore, the minimum ( k ) is ( frac{1}{2} ).**Final Answer**(1) The function ( f(x) = x ) satisfies the Lipschitz condition with ( k = 1 ).(2) The minimum value of ( k ) for ( f(x) = sqrt{x+1} ) is (boxed{frac{1}{2}}).

💡First, I recognize that both lines intersect at the point (4, 17). This means that when x is 4, y is 17 for both equations.I'll start with the first equation, y = 2mx + 5. Substituting x = 4 and y = 17 into this equation gives me:17 = 2m * 4 + 5Simplifying this, I get:17 = 8m + 5Subtracting 5 from both sides:12 = 8mDividing both sides by 8:m = 12 / 8 = 1.5Next, I'll use the second equation, y = 4x + b. Again, substituting x = 4 and y = 17:17 = 4 * 4 + bSimplifying this:17 = 16 + bSubtracting 16 from both sides:b = 1Finally, to find b + m, I add the values of b and m:b + m = 1 + 1.5 = 2.5

💡Okay, so I have this problem where I need to find all functions from the integers to the integers that satisfy the equation f(x + y) = f(x) + f(y) + 2 for all integers x and y. Hmm, functional equations can be tricky, but let me try to break this down step by step.First, I remember that functional equations often require substituting specific values for variables to simplify the equation and find patterns or properties of the function. Maybe I can start by plugging in some simple values for x and y.Let me try setting both x and y to 0. That should give me some information about f(0). So, if x = 0 and y = 0, the equation becomes:f(0 + 0) = f(0) + f(0) + 2Which simplifies to:f(0) = 2f(0) + 2Hmm, let's solve for f(0). Subtract 2f(0) from both sides:f(0) - 2f(0) = 2This simplifies to:-f(0) = 2So, multiplying both sides by -1:f(0) = -2Alright, that's a good start. Now I know that f(0) is -2. Let me see if I can find another substitution that might help. Maybe setting y = 1. Let's try that.If I set y = 1, the equation becomes:f(x + 1) = f(x) + f(1) + 2This looks like a recursive relation. It tells me that the value of the function at x + 1 is related to its value at x. Maybe I can use this to express f(x) in terms of f(0) or f(1).Let me think about how functions behave with such recursive relations. If I can express f(x) in terms of f(x - 1), I might be able to find a general form for f(x). Let's try to unravel this recursion.Starting from f(x + 1) = f(x) + f(1) + 2, I can write:f(x + 1) - f(x) = f(1) + 2This suggests that the difference between consecutive terms is constant, which means that f(x) is a linear function. In other words, f(x) should be of the form f(x) = ax + b, where a and b are constants.Let me test this assumption. Suppose f(x) = ax + b. Then, substituting into the original equation:f(x + y) = a(x + y) + bOn the other hand, f(x) + f(y) + 2 = (ax + b) + (ay + b) + 2 = a(x + y) + 2b + 2So, setting these equal:a(x + y) + b = a(x + y) + 2b + 2Subtracting a(x + y) from both sides:b = 2b + 2Solving for b:b - 2b = 2-b = 2b = -2So, the constant term b is -2. That matches what we found earlier for f(0). Therefore, the function must be of the form f(x) = ax - 2.Now, let's verify if this form satisfies the original functional equation. Substitute f(x) = ax - 2 into f(x + y):f(x + y) = a(x + y) - 2On the other hand, f(x) + f(y) + 2 = (ax - 2) + (ay - 2) + 2 = ax + ay - 4 + 2 = a(x + y) - 2So, both sides are equal, which means our assumption that f(x) is linear of the form ax - 2 is correct.Therefore, all functions satisfying the given equation are linear functions of the form f(x) = ax - 2, where a is an integer constant.I think that's it. I don't see any other possibilities, and the steps seem to check out. I started by finding f(0), then used a substitution to find a recursive relation, assumed a linear form, verified it, and confirmed that it works. So, I'm confident that this is the solution.**Final Answer**The functions satisfying the equation are all linear functions of the form boxed{f(x) = ax - 2} where ( a ) is an integer.

💡Okay, so I have this problem here: From the four prime numbers 3, 5, 7, and 11, if we take out two different numbers each time as a and b, the total number of different values obtained from lg a - lg b is... and then there are four options: A:6, B:8, C:12, D:16.Hmm, let me try to figure this out step by step. First, I know that lg a - lg b can be rewritten using logarithm properties. Specifically, lg a - lg b is equal to lg(a/b). So, instead of dealing with the difference of two logarithms, I can think of it as the logarithm of the ratio of a to b.Alright, so now the problem simplifies to finding how many different values of lg(a/b) we can get when a and b are distinct numbers from the set {3, 5, 7, 11}. Since a and b are different, each pair (a, b) will give a different ratio a/b, and thus a different logarithm value.Wait, but hold on. Is that necessarily true? Because if two different pairs (a, b) result in the same ratio a/b, then their logarithms would be the same. So, I need to check whether all the ratios a/b are unique or if some of them might repeat.Let me list all possible pairs of a and b. Since order matters here because a/b is different from b/a, each pair is unique. So, for four numbers, the number of ordered pairs where a ≠ b is 4 × 3 = 12. So, there are 12 possible ordered pairs.But does each of these 12 pairs give a unique ratio? Let me check.Let me list all possible a/b ratios:1. 3/52. 3/73. 3/114. 5/35. 5/76. 5/117. 7/38. 7/59. 7/1110. 11/311. 11/512. 11/7Now, I need to check if any of these ratios are equal. Let me see:- 3/5 is approximately 0.6- 3/7 is approximately 0.4286- 3/11 is approximately 0.2727- 5/3 is approximately 1.6667- 5/7 is approximately 0.7143- 5/11 is approximately 0.4545- 7/3 is approximately 2.3333- 7/5 is approximately 1.4- 7/11 is approximately 0.6364- 11/3 is approximately 3.6667- 11/5 is approximately 2.2- 11/7 is approximately 1.5714Looking at these decimal approximations, none of them seem to be equal. Each ratio is unique. Therefore, each of the 12 ordered pairs gives a distinct ratio, and thus each lg(a/b) will be a unique value.Wait, but just to be thorough, maybe some ratios could be equal if simplified? Let me check:- 3/5 cannot be simplified further.- 3/7 is also prime, so cannot be simplified.- 3/11 is the same.- 5/3 is the reciprocal of 3/5, so unless 3/5 equals 5/3, which it doesn't, they are different.- Similarly, 5/7 and 7/5 are reciprocals, so different.- 5/11 and 11/5 are reciprocals, different.- 7/3 and 3/7 are reciprocals, different.- 7/11 and 11/7 are reciprocals, different.- 11/3 is unique.- 11/5 is unique.So, indeed, all ratios are unique because all the primes are distinct and none of the fractions can be simplified to the same value. Therefore, each of the 12 ratios is unique, which means each lg(a/b) is unique.Therefore, the total number of different values obtained from lg a - lg b is 12.But wait, let me double-check. The problem says "different values obtained from lg a - lg b." So, if we have 12 different ratios, each with a unique logarithm, then we have 12 different values.But hold on, is there any chance that two different ratios could result in the same logarithm? For example, if a/b = c/d, then lg(a/b) = lg(c/d). But in our case, since all the ratios are unique, as we saw, their logarithms must also be unique because the logarithm function is injective (one-to-one). So, if a/b ≠ c/d, then lg(a/b) ≠ lg(c/d).Therefore, yes, all 12 values are unique.So, the answer should be 12, which is option C.But wait, let me think again. The problem says "take out two different numbers each time as a and b." So, does that mean that a and b are selected without considering order? Or is it considering order?Wait, in the problem statement, it's written as "take out two different numbers each time as a and b." So, it's not entirely clear whether order matters or not. If order doesn't matter, then the number of unordered pairs is C(4,2) = 6, which would give 6 different ratios. But in that case, each ratio would have a reciprocal, so the number of unique lg(a/b) would be 6, but considering that lg(a/b) = -lg(b/a), so each pair would give two different values, positive and negative. So, in that case, the number of different values would be 12 as well.Wait, this is confusing. Let me clarify.If the problem considers ordered pairs (a, b), then it's 12 different ordered pairs, each giving a unique ratio, hence 12 unique lg(a/b).If the problem considers unordered pairs {a, b}, then each pair can be arranged in two ways: a/b and b/a, which are reciprocals. So, for each unordered pair, we get two different values of lg(a/b): one positive and one negative. So, the number of unique values would still be 12 because each unordered pair contributes two unique logarithms.But wait, the problem says "take out two different numbers each time as a and b." So, it's not clear whether a and b are ordered or unordered. But in the context of subtraction, lg a - lg b is different from lg b - lg a, so it's likely that order matters.Therefore, the number of different values is 12.But let me check the options. The options are 6, 8, 12, 16. So, 12 is an option, which is C.Alternatively, if the problem had considered unordered pairs, then the number of unique ratios would be 6, but since each ratio and its reciprocal are different, the number of unique logarithms would still be 12 because each reciprocal would just be the negative of the original logarithm. So, even if you consider unordered pairs, the number of unique lg(a/b) values is 12.Wait, but hold on. If you consider unordered pairs, you have 6 pairs, each giving two logarithms: one positive and one negative. So, in total, 12 different values. So, regardless of whether you consider ordered or unordered pairs, the number of unique lg(a/b) values is 12.Therefore, the answer is C:12.But just to be absolutely sure, let me think about it another way. The number of possible ordered pairs is 12, each giving a unique ratio, hence 12 unique logarithms. So, yes, 12 is the correct answer.I think I've covered all the angles here. So, I'm confident that the answer is 12.**Final Answer**boxed{C}