answer：Alright, so I need to figure out which location had the highest snowfall difference compared to Mrs. Hilt's house. Okay, let's see. Mrs. Hilt's house got 29.7 inches of snow. Then there's Brecknock Elementary School with 17.3 inches, the town library with 23.8 inches, and the city park with 12.6 inches. First, I think I need to find out how much more or less snow each of these places got compared to Mrs. Hilt's house. So, I guess I need to subtract the snowfall at each location from Mrs. Hilt's snowfall. That should give me the difference.Let's start with Brecknock Elementary School. They had 17.3 inches. So, subtracting that from Mrs. Hilt's 29.7 inches: 29.7 minus 17.3. Hmm, 29.7 minus 17 is 12.7, and then minus 0.3 more would be 12.4 inches. Okay, so Brecknock Elementary School had 12.4 inches less snow than Mrs. Hilt's house.Next, the town library had 23.8 inches. Subtracting that from 29.7: 29.7 minus 23.8. Let's see, 29 minus 23 is 6, and 0.7 minus 0.8 is negative 0.1, so altogether that's 5.9 inches. So, the town library had 5.9 inches less snow than Mrs. Hilt's house.Now, the city park had 12.6 inches. Subtracting that from 29.7: 29.7 minus 12.6. That's 17.1 inches. So, the city park had 17.1 inches less snow than Mrs. Hilt's house.Wait, but the question is about the highest snowfall difference compared to Mrs. Hilt's house. So, I think that means which location had the biggest difference, whether more or less. But in this case, all the other locations had less snow than Mrs. Hilt's house. So, the biggest difference would be the largest number when subtracting.Looking at the differences: Brecknock Elementary School is 12.4 inches less, town library is 5.9 inches less, and city park is 17.1 inches less. So, 17.1 inches is the largest difference.But just to make sure, let me double-check my calculations. For Brecknock: 29.7 minus 17.3 equals 12.4. Yep, that's correct. Town library: 29.7 minus 23.8 equals 5.9. That seems right. City park: 29.7 minus 12.6 equals 17.1. Yeah, that's accurate.So, the city park had the highest snowfall difference compared to Mrs. Hilt's house, with a difference of 17.1 inches.

answer：Okay, so I have this problem where the general term of a sequence is given by ( a_n = frac{1}{4} + cosleft(frac{npi}{2}right) ). I need to find the sum of the first 2012 terms, denoted as ( S_{2012} ). The options are A: 1006, B: 2012, C: 503, D: 0.Hmm, let me start by understanding the sequence. The term ( cosleft(frac{npi}{2}right) ) suggests that the cosine function is periodic. Since the cosine function has a period of ( 2pi ), but here the argument is ( frac{npi}{2} ), so the period of this cosine term should be ( frac{2pi}{pi/2} = 4 ). That means every 4 terms, the cosine part will repeat its values.So, the sequence ( a_n ) is periodic with period 4. Let me write out the first few terms to see the pattern.For ( n = 1 ):( a_1 = frac{1}{4} + cosleft(frac{pi}{2}right) = frac{1}{4} + 0 = frac{1}{4} ).For ( n = 2 ):( a_2 = frac{1}{4} + cosleft(piright) = frac{1}{4} + (-1) = -frac{3}{4} ).For ( n = 3 ):( a_3 = frac{1}{4} + cosleft(frac{3pi}{2}right) = frac{1}{4} + 0 = frac{1}{4} ).For ( n = 4 ):( a_4 = frac{1}{4} + cos(2pi) = frac{1}{4} + 1 = frac{5}{4} ).So, the first four terms are ( frac{1}{4}, -frac{3}{4}, frac{1}{4}, frac{5}{4} ). Let me add these up to see the sum of one period.Sum of first four terms:( frac{1}{4} + (-frac{3}{4}) + frac{1}{4} + frac{5}{4} ).Calculating step by step:- ( frac{1}{4} - frac{3}{4} = -frac{2}{4} = -frac{1}{2} )- ( -frac{1}{2} + frac{1}{4} = -frac{1}{4} )- ( -frac{1}{4} + frac{5}{4} = frac{4}{4} = 1 )So, the sum of each block of 4 terms is 1. That's interesting. Now, since the sequence is periodic with period 4, every set of 4 terms will add up to 1.Now, I need to find ( S_{2012} ), which is the sum of the first 2012 terms. Since 2012 is a multiple of 4, I can divide 2012 by 4 to find how many such blocks there are.Calculating ( 2012 div 4 ):( 2012 div 4 = 503 ).So, there are 503 blocks of 4 terms each. Since each block sums to 1, the total sum ( S_{2012} ) should be ( 503 times 1 = 503 ).Let me double-check my calculations to make sure I didn't make a mistake.First, verifying the periodicity: The cosine term ( cosleft(frac{npi}{2}right) ) indeed has a period of 4 because ( frac{(n+4)pi}{2} = frac{npi}{2} + 2pi ), and cosine has a period of ( 2pi ). So, every 4 terms, the cosine part repeats.Calculating the first four terms again:- ( a_1 = frac{1}{4} + 0 = frac{1}{4} )- ( a_2 = frac{1}{4} - 1 = -frac{3}{4} )- ( a_3 = frac{1}{4} + 0 = frac{1}{4} )- ( a_4 = frac{1}{4} + 1 = frac{5}{4} )Sum:( frac{1}{4} - frac{3}{4} + frac{1}{4} + frac{5}{4} = (frac{1}{4} + frac{1}{4}) + (-frac{3}{4} + frac{5}{4}) = frac{2}{4} + frac{2}{4} = 1 ). Yep, that's correct.Number of blocks:( 2012 div 4 = 503 ). That's straightforward.Total sum:( 503 times 1 = 503 ). So, the answer should be 503, which is option C.I don't see any mistakes in my reasoning. The key was recognizing the periodicity of the cosine term, calculating the sum over one period, and then determining how many such periods fit into 2012 terms. Since 2012 is perfectly divisible by 4, there are no leftover terms to consider, which simplifies the calculation.Just to be thorough, let me consider what would happen if the number of terms wasn't a multiple of 4. For example, if I had 2013 terms, I would have 503 full blocks (summing to 503) plus one additional term, which would be ( a_{2013} = frac{1}{4} + cosleft(frac{2013pi}{2}right) ). Since ( 2013 mod 4 = 1 ), this would be ( frac{1}{4} + 0 = frac{1}{4} ). So, the total sum would be ( 503 + frac{1}{4} = 503.25 ). But in this case, since we have exactly 2012 terms, which is 503 blocks, the sum is just 503.Another way to think about it is to separate the sequence into two parts: the constant term ( frac{1}{4} ) and the cosine term. The sum ( S_n ) can be written as:( S_n = sum_{k=1}^{n} left( frac{1}{4} + cosleft( frac{kpi}{2} right) right) = frac{n}{4} + sum_{k=1}^{n} cosleft( frac{kpi}{2} right) ).So, the sum is the sum of the constants plus the sum of the cosine terms. The constant sum is straightforward: ( frac{n}{4} ). The cosine sum is more complex, but since it's periodic with period 4, we can analyze it similarly.Looking at the cosine terms:( cosleft( frac{pi}{2} right) = 0 ),( cos(pi) = -1 ),( cosleft( frac{3pi}{2} right) = 0 ),( cos(2pi) = 1 ),and then it repeats.So, over each period of 4 terms, the cosine terms sum to ( 0 - 1 + 0 + 1 = 0 ). Therefore, the sum of the cosine terms over any multiple of 4 terms is zero. Hence, the total sum ( S_n ) is just ( frac{n}{4} ) when ( n ) is a multiple of 4.Wait, hold on, that contradicts my earlier conclusion where I found the sum of four terms to be 1. Let me check this.If I separate the sum into constants and cosine terms:( S_n = sum_{k=1}^{n} frac{1}{4} + sum_{k=1}^{n} cosleft( frac{kpi}{2} right) = frac{n}{4} + sum_{k=1}^{n} cosleft( frac{kpi}{2} right) ).But earlier, I found that the sum of four terms is 1, which is ( frac{4}{4} + 0 = 1 ). So, in that case, the cosine sum over four terms is 0, which is consistent.Wait, but in my initial calculation, the sum of four terms was 1, which is equal to ( frac{4}{4} + 0 = 1 ). So, that makes sense.But in the case where ( n ) is not a multiple of 4, the cosine sum won't be zero. For example, if ( n = 5 ), the cosine sum would be ( 0 - 1 + 0 + 1 + 0 = 0 ), but the constant term would be ( frac{5}{4} ). So, the total sum would be ( frac{5}{4} + 0 = frac{5}{4} ).Wait, but if ( n = 5 ), the sum of the first five terms would be:( a_1 + a_2 + a_3 + a_4 + a_5 = frac{1}{4} - frac{3}{4} + frac{1}{4} + frac{5}{4} + frac{1}{4} ).Calculating step by step:- ( frac{1}{4} - frac{3}{4} = -frac{1}{2} )- ( -frac{1}{2} + frac{1}{4} = -frac{1}{4} )- ( -frac{1}{4} + frac{5}{4} = 1 )- ( 1 + frac{1}{4} = frac{5}{4} )So, that's consistent with the separated sum.Therefore, in general, for ( n ) terms, ( S_n = frac{n}{4} + sum_{k=1}^{n} cosleft( frac{kpi}{2} right) ). Since the cosine sum over each period is zero, for ( n ) being a multiple of 4, the cosine sum is zero, so ( S_n = frac{n}{4} ).Wait, but in the initial four terms, the sum was 1, which is ( frac{4}{4} = 1 ). So, that's consistent.But in my first approach, I considered the sum of four terms as 1, which is ( frac{4}{4} + 0 = 1 ). So, that's correct.Therefore, for ( n = 2012 ), which is a multiple of 4, ( S_{2012} = frac{2012}{4} = 503 ).So, that's another way to see it, by separating the sum into the constant part and the cosine part. Since the cosine part cancels out over each period, the total sum is just the sum of the constants, which is ( frac{n}{4} ).Therefore, the answer is 503, which is option C.I think both approaches confirm that the answer is 503. I don't see any flaws in my reasoning. It's important to recognize the periodicity of the cosine term and how it affects the sum. By breaking down the problem into manageable parts and verifying each step, I can be confident in the solution.Another thing to consider is whether the periodicity affects the sum in any other way, but since the cosine terms cancel out over each period, it doesn't complicate the sum beyond what's already considered. So, the key insight is recognizing the period and how it simplifies the summation.In summary, by analyzing the periodicity of the cosine term, calculating the sum over one period, and then extending that to the total number of terms, I can efficiently find the sum without having to compute each term individually. This approach saves a lot of time, especially for large ( n ) like 2012.I think I've covered all the necessary steps and checked my work thoroughly. I feel confident that 503 is the correct answer.**Final Answer**boxed{C}

answer：First, I need to identify the complex numbers ( z_1 ) and ( z_2 ) based on the given information. The point corresponding to ( z_1 ) is ( (2, 3) ), which means ( z_1 = 2 + 3i ). The complex number ( z_2 ) is given as ( -1 + 2i ).Next, I'll subtract ( z_2 ) from ( z_1 ) to find ( z ):[z = z_1 - z_2 = (2 + 3i) - (-1 + 2i)]Simplifying the expression:[z = 2 + 3i + 1 - 2i = 3 + i]The resulting complex number ( z = 3 + i ) corresponds to the point ( (3, 1) ) in the complex plane.Since both the real part (3) and the imaginary part (1) are positive, the point ( (3, 1) ) lies in the first quadrant of the complex plane.

answer：Okay, so I have this problem where there's a half-circle sector with a radius of 6 inches, and when you roll it up, it forms a right circular cone. I need to find the volume of that cone. Hmm, let's see. I remember that when you roll a sector into a cone, the radius of the sector becomes the slant height of the cone. So, the slant height (l) of the cone is 6 inches.Now, the circumference of the base of the cone is equal to the length of the arc of the sector. Since it's a half-circle, the arc length is half of the circumference of the full circle. The circumference of a full circle with radius 6 inches is 2π*6 = 12π inches. So, half of that is 6π inches. That means the circumference of the base of the cone is 6π inches.From the circumference, I can find the radius of the base of the cone. The formula for circumference is 2πr, so if 2πr = 6π, then r = 3 inches. Okay, so the radius of the base of the cone is 3 inches.Now, I need to find the height of the cone to calculate the volume. I know the slant height (l) is 6 inches, and the radius (r) is 3 inches. I remember the Pythagorean theorem for cones: l² = r² + h². Plugging in the values, 6² = 3² + h², which simplifies to 36 = 9 + h². Subtracting 9 from both sides, I get h² = 27. Taking the square root of both sides, h = √27, which simplifies to 3√3 inches.Finally, the volume of a cone is given by (1/3)πr²h. Plugging in the values, that's (1/3)π*(3)²*(3√3). Calculating that, (1/3)*π*9*3√3. The 1/3 and 9 cancel out to 3, so it's 3*3√3*π, which is 9√3π. So, the volume is 9π√3 cubic inches.

answer：To find the sum of the seven smallest distinct positive integer multiples of 12, I will first list these multiples. The multiples are 12, 24, 36, 48, 60, 72, and 84.Next, I will add these numbers together step by step:- Start with 12.- Add 24 to get 36.- Add 36 to get 72.- Add 48 to get 120.- Add 60 to get 180.- Add 72 to get 252.- Finally, add 84 to reach a total sum of 336.Therefore, the sum of the seven smallest distinct positive integer multiples of 12 is 336.

answer：Okay, so John bought a tennis racket, some sneakers, and a sports outfit. The sneakers cost 200 and the sports outfit cost 250. Altogether, he spent 750. I need to find out how much the tennis racket cost.First, I should figure out how much John spent on the sneakers and the sports outfit combined. That means adding the cost of the sneakers and the sports outfit together. So, 200 for the sneakers plus 250 for the sports outfit. Let me do that math: 200 plus 250 equals 450. So, he spent 450 on the sneakers and the sports outfit.Now, I know that the total amount he spent on all three items was 750. If I subtract the amount he spent on the sneakers and the sports outfit from the total amount, that should give me the cost of the tennis racket. So, 750 minus 450 equals... let me calculate that. 750 minus 450 is 300. Wait, does that make sense? Let me double-check my calculations. 200 plus 250 is definitely 450. And 750 minus 450 is 300. Yeah, that seems right. So, the tennis racket must have cost 300.I think that's the answer. The price of the racket is 300.